Journal of Chromatography A, 1399 (2015) 88–93
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Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Short communication
Accuracy verification of surface energy components measured by inverse gas chromatography Mohammad Amin Mohammad a,b,∗ a b
Department of Pharmacy and Pharmacology, University of Bath, Bath BA2 7AY, UK Faculty of Pharmacy, University of Damascus, Damascus, Syria
a r t i c l e
i n f o
Article history: Received 21 January 2015 Received in revised form 22 April 2015 Accepted 22 April 2015 Available online 30 April 2015 Keywords: Accuracy verification criteria Adhesion retention factor Chromatographic adhesion law Inverse gas chromatography Surface components
a b s t r a c t Inverse gas chromatography (IGC) measures the retention times of probes which are then used to calculate the surface properties of solids. No method is available to verify how much the measured values are close to their accurate values. According to the chromatographic adhesion law, the accurate determination of a the dispersive retention factor (KCH ) is a necessary prerequisite to obtain accurate surface components. 2
Employing two equations in this paper, %Sd , which is the percentage deviation of dispersive component d (Sd ) from its accurate value (S0 ), was correlated firstly to %CVln K a , the percentage coefficient of variation CH2
a a of ln KCH , and secondly to FEKaCH2 , the factor error of KCH , via two linear equations. The first equation is to 2
2
d , and the second equation is to estimate outline the upper and lower limits of the uncertainty range of S0 d d S0 . To minimize the uncertainty range of S0 to less than ±5%, %CVln K a should be less than 0.7%. Then CH2
considering the sign of FEKaCH2 narrows the uncertainty range to either its upper or lower half. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Inverse gas chromatography (IGC) is a simple, fast and accurate technique for characterizing the surface properties of solids in any form [1]. IGC is used to evaluate the dispersive surface energy (Sd ) of the solids using either Dorris–Gray [2] or Schultz [3] calculation methods which are based on the net retention volumes (Vn ) of n-alkane probes at infinite dilution. The electron acceptor (S+ ) and the electron donor (S− ) components of the solids are then calculated by subtracting the dispersive component from the total free energy of adsorption using the van Oss–Good–Chaudhury method [4]. These methods have been recently reviewed in details [1]. To enhance the accuracy of the obtained values, various researchers recently scrutinized the calculation methods. Shi et al. found that the values of Sd vary according to the calculation methods, and then they normalized this variation by determining the ratio between Dorris–Gray and Schultz methods [5]. It was also found that the calculated surface components of the same sample vary up to 16% depending on the used n-alkane series, and they attributed this discrepancy to uneven differences in the literature parameters of n-alkanes which are their dispersive free energy (ld )
∗ Department of Pharmacy and Pharmacology, University of Bath, Bath BA2 7AY, UK. Tel.: +44 1225 386797. E-mail address: [email protected] http://dx.doi.org/10.1016/j.chroma.2015.04.044 0021-9673/© 2015 Elsevier B.V. All rights reserved.
and cross-sectional area (˛) [6]. The traditional literature parameters of n-alkanes listed in the papers using Schultz method are not accurate enough [7] and Shi et al. reported [5] that they are not in good agreement with the same parameters in solvents handbook. To overcome calculated IGC data variability, Mohammad combined the van Oss–Good–Chaudhury concept, the Dorris–Gray equation, the Schultz equation, the Fowkes equation and group contribution theory to establish the chromatographic adhesion law [8]. This law a ), the electron acceptor uses the dispersive retention factor (KCH 2 a ), and the electron donor retention factor (K a ) retention factor (Kl+ l− a and K a valto calculate and characterise the surface properties. Kl+ l− a , and they are calculated from the net retention ues depends on KCH 2 times of the probes (tn ) obtained from the measured retention times of the probes (tr ). Chromatographic apparatus has been designed in order to obtain measurements of high precision. However, random errors can be minimized but not eliminated [9]. The tolerances and the failures of the equipment used for the control of the temperature and flow rate, faulty operation, and sampling procedure cause variations of column temperature and carrier gas flow rate which, in turn, cause significant changes in tr s [10,11]. tr s also change with variation in column performance or column overloading with sample [12]. These potential variations in tr s of a probe or many probes a , and so the calculated surface properties will change tn , Vn and KCH 2 deviate from their accurate values.
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
Researchers have published several hundred papers on surface characterization with IGC. Their criterion to judge the accuracy of IGC measurements for tr s is the linearity of the n-alkane line “r”, 0.5
which is the plot of RTlnVn versus ˛(ld ) . Usually, “r” should be ≥0.9995 [13]. To date, no method has been reported to verify the accuracy of IGC measurement for tr s, and to evaluate how tr s’ inaccuracy affects the accuracy of obtained surface properties. In a these studies, the dependence of Sd , S− and S+ on KCH is clarified 2 a and then KCH is used to establish the accuracy verification criteria 2 of IGC measurements. These criteria will be used to evaluate the d ). deviation of obtained Sd from its accurate value (S0 2. Discussion 2.1. Mathematical derivation The chromatographic adhesion law can be used to calculate Sd , + S and S− , and its general equation is: Kia
E a /kT
=e
i
(1)
Kia
Eia
is the adhesion retention factor of a chemical group, where represents the increased adhesion energy due to this group, k is the Boltzmann constant, and T is the column temperature (in degrees Kelvin). CH2 participates in the chromatographic adhesion via its dispera is calculated from tn s of any two sive component only, and its KCH 2 successive n-alkanes, as follows: tnci+1 tnci
the retention time of its theoretical n-alkane reference (tnl−,ref ), as follows: a Kl− =
ln tnl−,ref =
a El− = 2al− (l− )
a ECH = 2
(3)
In this case, the general equation becomes as follows: E a
a KCH =e
CH2
/kT
(4)
2
a Kl+
of a monopolar electron acceptor probe (l+) (e.g., dichloromethane or chloroform) can be calculated from its net retention time (tnl+ ) and the retention time of its theoretical nalkane reference (tnl+,ref ), as follows: a Kl+ =
tnl+ tnl+,ref
(5)
and tnl+,ref can be calculated as follows:
a ln tnl+,ref = ln tnci + ln KCH
2
d ) ˛l+ (l+
0.5
d ) ˛ci+1 (l,ci+1
d ) − ˛ci (l,ci
0.5
0.5
d ) − ˛ci (l,ci
0.5
a El+ = 2al+ (l+ )
0.5
(s− )
0.5
Sd =
E a /kT
a Kl+ =e
l+
(s+ )
d ) − ˛ci (l,ci
0.5
0.5
(10)
(11)
(12)
l−
2
˛2CH CH2 a )2 0.477 (T ln Kl+
a2l+ l+
a )2 0.477 (T ln Kl−
(mJ m−2 )
(13)
a2l− l−
(mJ m−2 )
(14)
(mJ m−2 )
(15)
where the units of ˛ are Å2 and of are mJ m−2 in all equations. Rounding off 0.477 to the nearest tenth (to be 0.5) will cause 4.5% increase in calculated surface components, and rounding off 0.477 to two decimal digits (to be 0.48) will cause 0.6% increase in calculated surface components. Eqs. (6), (10), (13)–(15) clarify that all surface components of a . Therefore, solids (Sd , S− and S+ ) depend on the value of KCH 2 a the accuracy of the measured value of KCH is a prerequisite for 2
verifying the accuracy of the calculated values of Sd , S− and S+ . Supposedly, five n-alkanes (n-hexane to n-decane) are used to a can be calculated from the net probe a solid. In this case, KCH 2 retention times of any two successive n-alkanes, as follows: =
tn7 tn6
(16)
=
tn8 tn7
(17)
=
tn9 tn8
(18)
2
CHC8,C7 2
Ka
CHC9,C8 2
Ka
CHC10,C9 2
=
tn10 tn9
(19)
tn is calculated as follows: tn = (tr − t0 )
a of a monopolar electron donor probe (l−) (e.g., ethyl acetate Kl− or toluene) can be calculated from its net retention time (tnl− ) and
0.5
a )2 0.477 (T ln KCH
Ka
(8)
0.5
0.5
2
(6)
In this case, the general equation becomes as follows:
d ) ˛ci+1 (l,ci+1
d ) − ˛ci (l,ci
Combining Eqs. (3), (7) and (11) with Eqs. (4), (8) and (12), respectively, produces the following similar equations to calculate the three surface components of the solids:
CHC7,C6
(7)
0.5
E a /kT
a Kl− =e
Ka
d and a are the dispersive component of l+ and its where l+ l+ cross-sectional area. As the dispersive component of l+ is omitted by its theoretical n-alkane, l+ contributes to the chromatographic adhesion energy via its electron acceptor component (l+ ), and its a ) is calculated as follows: additive adhesion energy (El+
d ) ˛l− (l−
In this case, the general equation becomes as follows:
s+ =
0.5 2˛CH2 (CH2 )0.5 (Sd )
a ln tnci + ln KCH 2
d and a are the dispersive component of l− and its crosswhere l− l− sectional area. As the dispersive component of l− is omitted by its theoretical n-alkane, l− contributes to the chromatographic adhesion energy via its electron donor component (l− ) only, and its a ) is calculated as follows: additive adhesion energy (El−
The additive contribution of CH2 to the adhesion energy is only a ) and it is calculated as follows: dispersive (ECH 2
(9)
s− =
2
tnl− tnl−,ref
tnl−,ref can be calculated as follows:
(2)
a KCH =
89
(20)
where t0 is the retention time of a non-adsorbing marker. Combining the logarithms of any combination of Eqs. (16)–(19) a yields that ln KCH can be calculated from the subtraction of the 2 logarithms of the net retention times (lntn ) of any two n-alkanes
90
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
divided by the subtraction of their carbon atom numbers (n), which can be translated to the following equation: a ln KCH = 2
ln tn n
(21)
Integration of Eq. (21) gives: (22)
2
2.2. IGC experiment
The slope of Eq. (22), which is the same as that of Conder–Young a [14], averages out ln KCH from the n-alkane line (lntn vs. n). 2 Ideally, if trC6 , trC7 , trC8 , trC9 and trC10 are measured accurately, the standard deviation of the slope of Eq. (22) (SDln K a ) and its CH2
percentage coefficient of variation (%CVln K a
CH2
) are zero. Practically,
they have positive values, because there are experimental errors in tr s’ measurements as set out in the introduction. %CVln K a is
CH2
%CVln K a
CH2
SDln Ka
CH2
=
× 100
a ln KCH
(23)
CH2
increases, the possible deviation of Sd from its
d ) increases, and the percentage deviation of d accurate value (S0 S d from S0 is:
d ) (Sd − S0
× 100
d S0
(24)
CH2
a be used to determine the sign of %Sd . The factor error of KCH
2
(FEKaCH ), which has zero (ideally), positive or negative values, and 2
so it would be correlated with the sign of %Sd . FEKaCH was derived 2 from Eqs. (16) to (19) as follows, if tr s was determined accurately: C7,C6 2
CH
ln K a
= Ka
= Ka
C8,C7 2
CH
C7,C6 2
CH
(ln K a
C7,C6 CH 2
C9,C8 2
CH
= ln K a
C8,C7 2
CH
+ ln K a
C8,C7 CH 2
= Ka
= ln K a
C9,C8 2
CH
− ln K a
C9,C8 CH 2
= ln K a
− ln K a
C10,C9 CH 2
+ ln K a C8,C7 CH
− ln K a
2
CH
C9,C8 2
− ln K a
CH
C10,C9 2
FEKaCH and %Sd are zero. Therefore, the obtained Sd can be con-
)
Combining Eqs. (16)–(19) with Eq. (28) produces: = 100 × ln
In general FEKaCH 2
= 100 × ln
(tnCi )2 tnCi−2 tnCi+2
0.5
) × 100 = 0 (27)
(28)
(tnC8 )2 tnC6 tnC10
0.5
d d ) ) − ˛ci (l,ci of six n-alkanes (from n-pentane to of ˛ci+1 (l,ci+1 n-decane) listed in the Schultz and co-workers paper [3]. However, according to Dorris–Gray [2], ˛CH2 = 6 A˚ 2 and CH2 is calculated as follows:
The laboratory experiment was repeated until we obtained retention times of the five n-alkanes “r” = 1.00000, and %CVln K a ,
2
and (22), respectively. Then Sd was calculated using Eq. (13) of the chromatographic adhesion law. Also, Sd was calculated according to the Schultz and Dorris-Gray methods for reference. In Eq. (13), we set ˛CH2 (CH2 )0.5 equal to 35.72 A˚ 2 (mJ/m2 )0.5 , which is the average
(26)
C10,C9 2
CH
× 100 = FEKaCH
FEKaCH 2
where n is the carbon number of the homologous n-alkanes, and A and B are constants. This is because the linear relationship between ln(tr − t0 ) and n under isothermal conditions (Conder–Young concept) has been widely used (since the 1950s) to calculate t0 [14]. a using Eqs. (20) t0 was then used to calculate tn s, and ln KCH
CH2 = 35.6 + 0.058(293 − T ) (mJ m−2 )
2
2
(31)
(25)
C10,C9 2
CH
The left-hand side of Eq. (27) is zero in an ideal state. However, practically, the left-hand side can be zero, positive or negative, and considered as FEKaCH , so Eq. (27) becomes: (ln K a C7,C6 CH
ln(tr − t0 ) = An + B
2
%Sd can be zero (ideally), positive or negative, and the positive d towards higher values, and vice %Sd means Sd deviates from S0 versa. However, %CVln K a is either zero or positive, so it cannot
Ka
Experiments were performed using inverse gas chromatography (IGC 2000, Surface Measurement Systems Ltd., UK). Lactose monohydrate (Pharmatose 200M, Fonterra Ltd., New Zealand) (63–90 m) was packed into a pre-silanised glass column (300 mm × 3 mm i.d.). The column was analysed at 30 ◦ C and zero relative humidity, using anhydrous helium gas as the carrier. A series of n-alkanes (n-hexane to n-decane) was injected at the infinite dilution region. Their retention times followed from detection using a flame ionization detector (FID). t0 is considered to be the value that gives the best linearity in the Conder–Young equation:
2
When %CVln K a
%Sd =
2
%Sd .
a ln tn = ln KCH n + constant
calculated as follows:
Eq. (27) and therefore Eqs. (28)–(30) have six alternative forms as the four (equal) components within Eq. (27), when the total is set to = 0 requires two components to be added and two to be subtracted. The simulated experiments (Section 2.2 discussed below) show that FEKaCH , calculated using Eq. (30), is well correlated with
(29)
(30)
(32)
CH2
2
d which was 47.0 mJ m−2 as calculated by both Eq. (13) sidered as S0 of the chromatographic adhesion law and the Schultz method. Supposedly, one or more of tr s deviate during IGC measurements due to practical reasons or random errors. Therefore, tr s of the five n-alkanes would be shift up (+), down (−) or stay accurate (0). In this case, the number of possibilities is equal to 35 = 243; these possibilities are given codes starting from 1 until 243 (Supplementary data, Table 1S). Code 1 represents the original accurate tr s. Code 2 and code 3 represent another accurate sets of numbers, as tr s of all n-alkanes were shifted down and up simultaneously at the same rate (i.e., changing the flow rate), respectively. Thered . fore, simulated experiments with codes 1–3 generate the same S0 For other possibilities, we created 50 simulated experiments (50 sets of tr s) spreading over the traditional acceptable range of “r” [1–0.9995]. However, each set of tr s that produced negative t0 was omitted. The total number of experiments generated was 11,385 simulated experiments. For each experiment, Sd (using chromatoa , graphic adhesion law, Schultz, and Dorris-Gray methods), KCH 2
d , Again, if tr s of all n-alkanes is determined accurately, Sd is S0 a d and %CVln K a , FEKCH and %S are zero, also the linear fit degree CH2
2
for the plot of RTlnVn versus ˛(Ld )
0.5
becomes exactly one.
%CVln K a
CH2
, FEKaCH , %Sd , and “r” were calculated (Supplementary 2
data, Table 2S). Fig. 1 shows that Sd s calculated by both chromatographic adhesion law and Schultz method are similar and lower than those
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
91
0.5
Fig. 1. The possibilities of the dispersive surface energy values (Sd ) of lactose monohydrate as function of the linear fit degree “r” for the plot of RTlnVn versus ˛(Ld ) ; the dispersive surface energy values calculated by chromatographic adhesion law (blue), Schultz method (red) and Dorris-Gray method (green) (For interpretation of the color information in this figure legend, the reader is referred to the web version of the article.).
Fig. 2. The possibilities of the dispersive surface energy values (Sd ) of lactose monohydrate as function of the percentage coefficient of variation of dispersive retention factor (%CVln K a ); the dispersive surface energy values calculated by chromatographic adhesion law (blue), Schultz method (red) and Dorris-Gray method (green) (For CH2
interpretation of the color information in this figure legend, the reader is referred to the web version of the article.).
calculated by the Dorris–Gray method, and this observation is consistent with previous findings [5,8,15]. Also it shows that, irrespective of calculation methods, (Sd × 100)/(100 + 7.5%CVln K a ) CH2
d . When “r” approaches one, d varies by about ±15% from S0 S d . However, the possibilities of d values spread approaches S0 S widely (to lower and higher values) as “r” decreases from 1 toward 0.9995. Therefore, r ≥ 0.9995 is not a suitable criterion to accept d is IGC measurements because the uncertainty about Sd to be S0 high even at “r” close to one. However, Fig. 2 shows that %CVln K a
%CVln K a
CH2
(Fig. 3). Each value of %CVln K a
CH2
%Sd
is equivalent to a range
values spreading symmetrically about its zero value, the of distance between the upper and lower limits of the uncertainty (%upper Sd and %lower Sd ) is the uncertainty range about Sd to be d , this uncertainty range is outlined by two straight lines in Fig. 3 S0 which shows that %CVln K a linearly correlates with %upper Sd and CH2
%lower Sd with the following equation:
CH2
is a better criterion for the accuracy verification of IGC experid . The upper ments. At %CVln K a being zero or close to zero, Sd is S0 CH2
d increase and lower limits of the uncertainty about Sd to be S0 d linearly as functions of %CVln K a . As %S represents the percent-
age deviation of
Sd
CH2
from
d , S0
%Sd
was plotted as a function of
%limits Sd = ±7.5%CVln K a
(33)
CH2
The coefficient of determination “r2 of Eq. (33) is 1.000. This d of an investigated solid is with the means that we know that S0
92
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
d Fig. 3. The correlation between the limits of the percentage deviation of dispersive surface energy from its accurate value (the uncertainty range of S0 ) and the percentage coefficient of variation of dispersive retention factor (%CVln K a ). CH2
uncertainty range from (Sd × 100)/(100 + 7.5%CVln K a 100)/(100 − 7.5%CVln K E
CH2
) to (Sd ×
), and its equation can be represented as:
variables, all points were linearly regressed within the following linear equation:
CH2
%Sd = 0.3FEKaCH
(35)
2
d Uncertainty range of S0 = [(Sd × 100)/(100 + 7.5%CVln K a
CH2
(Sd × 100)/(100 − 7.5%CVln K a
CH2
) to
The correlation coefficient “r” of Eq. (35) is 0.88. This means that we can estimate that:
)] (34) d S0 =
(Sd × 100)
(36)
(100 + 0.35FEKaCH ) 2
If %CVln K a
CH2
d . The uncertainty equals zero, we know that Sd is S0
d increases as %CV range about S0 ln K a
CH2
increases, and vice versa.
d within We developed a mathematical model to estimate S0 its uncertainty range. FEKaCH was the best model we found. Fig. 4
Moreover, the sign of FEKaCH is the same as that of %Sd . This 2 helps to narrow the uncertainty range to either its upper or lower half, i.e., if FEKaCH is positive, Eq. (34) becomes: 2
2
shows the strong and direct correlation between FEKaCH and %Sd . 2 Most of the points form straight lines, but a few points form curves. Therefore, to approximate the correlation between these two
d Uncertainty range of S0
=
(Sd × 100)
(100 + 7.5%CVln K a
CH2
to (Sd ) )
(37)
Fig. 4. The correlation between the percentage deviation of dispersive surface energy from its accurate value (%Sd ) and the factor error of dispersive retention factor (FEKaCH ). 2
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
93
Table 1 d ) and its estimated value for three different powders; A is lactose monohydrate, B is micronized lactose, The uncertainty range of the accurate dispersive surface energy (S0 and C is recrystallized lactose. Solid
trC6 (min)
trC7 (min)
trC8 (min)
trC9 (min)
trC10 (min)
%CVln K a
FEKaCH
Sd (mJ m−2 )
Uncertainty range d (mJ m−2 ) of S0
Narrowed uncertainty range of d (mJ m−2 ) by considering S0 the sign of FEKaCH
d Estimated S0 (mJ m−2 )
A B C A*
0.561 0.660 2.716 0.602
1.130 1.271 3.178 1.168
3.128 3.324 4.336 3.045
9.104 10.135 7.562 9.102
28.673 33.102 16.357 28.392
0.908 0.354 0.679 0.689
14.5 4.8 1.5 −6.8
48.9 51.1 34.6 45.6
45.7–52.4 49.8–52.5 32.9–36.4 43.4–48.1
45.7–48.9 49.8–51.1 32.9–34.6 45.6–48.1
46.8 50.4 34.4 46.6
CH2
2
2
*
The experiment of material A was repeated to get lower %CVln K a
CH2
.
and if FEKaCH is negative, Eq. (34) becomes: 2
d = Uncertainty range of S0
(Sd ) to
(Sd × 100)
(100 − 7.5%CVln K a )
(38)
CH2
%CVln K a
CH2
Acknowledgments
, FEKaCH , and %Sd are ratios (unit less) and 2
a , d , d or/and t s vary due to varying solids, unchangeable if KCH r S0 S 2 temperature, flow rate, calculation methods or/and experimental errors. Therefore, the above simulated experiments (∼12,000) on one solid are enough to verify our suggested criteria of the accuracy of IGC measurements. Table 1 shows some examples of data and the application of the d different from new criteria. The data show that powder C has S0 those of both powders A and B because its uncertainty range does not overlap with others’ uncertainty ranges. However, the uncertainty ranges of A and B overlap partially. Therefore, we repeated the IGC experiment for A to minimize the uncertainty range by d valdecreasing %CVln K a . As a result, the difference between S0 CH2
ues of A and B becomes clear as concluded from their separated uncertainty ranges (last row of Table 1). Also, the narrowed uncertainty ranges obtained by considering the sign of FEKaCH show the 2 differences between the three solids, even before retesting solid d A. Moreover, the values of estimated S0 quantify the difference between the solids’ dispersive surface components. All estimated d using Eq. (36) values are within the narrowed uncertainty S0 ranges calculated using Eqs. (37) and (38). This elucidates the harmony between Eqs. (36) and (34) and its narrowed forms, Eqs. (37) and (38). Three or four n-alkanes can be used to outline the uncertainty d as to calculate range, but they are not enough to estimate S0 a FEKCH five n-alkanes are required. Also, six n-alkanes are more 2 than enough. 3. Conclusions d , but it is possible to be Practically, it is difficult to measure S0 close to its accurate value. It is essential to have criteria to quantify the uncertainty about Sd . %CVln Ka has been used to outline the CH2
d . This uncertainty range uncertainty range, and FEKaCH estimates S0 2 can be narrowed by repeating the experiment to obtain lower values of %CVln Ka . We recommend IGC experiments with %CVln K a CH2
CH2
less than 0.7% to minimize the uncertainty about measured Sd to less than ±5%. For example, when %CVln K a is 0.7% and Sd is CH2
50 mJ m−2 , we know that the accurate value of Sd is within the range of 47.5–52.8 mJ m−2 . Lowering %CVln K a below 0.7% minimizes CH2
factor is a necessary prerequisite to obtain accurate surface components. The new criteria have been applied herein to verify the accuracy of the dispersive component.
this uncertainty range. Moreover, considering the sign of FEKaCH 2 narrows the uncertainty range by half. The method reported herein highlights that the accuracy of the measured dispersive retention
I gratefully acknowledge University of Bath and CARA (Zeid AlBayaty and Kate Robertson) for providing an academic fellowship, and thank Dr. Ian S. Blagbrough (University of Bath), Prof James H. Davenport (University of Bath), and Prof Victor Rivero (visiting professor, University of Bath and Jalisco Institution, Mexico) for their helpful discussions. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.chroma.2015.04. 044 References [1] S. Mohammadi-Jam, K.E. Waters, Inverse gas chromatography applications: a review, Adv. Colloid Interface Sci. 212 (2014) 21–44. [2] G.M. Dorris, D.G. Gray, Adsorption of n-alkanes at zero surface coverage on cellulose paper and wood fibers, J. Colloid Interface Sci. 77 (1980) 353–362. [3] J. Schultz, L. Lavielle, C. Martin, The role of the interface in carbon fibre-epoxy composites, J. Adhesion 23 (1987) 45–60. [4] C. van Oss, R. Good, M. Chaudhury, Additive and nonadditive surface tension components and the interpretation of contact angles, Langmuir 4 (1988) 884–891. [5] B. Shi, Y. Wang, L. Jia, Comparison of Dorris-Gray and Schultz methods for the calculation of surface dispersive free energy by inverse gas chromatography, J. Chromatogr. A 1218 (2011) 860–862. [6] M.A. Mohammad, I.M. Grimsey, R.T. Forbes, An approach to normalise inverse gas chromatography data measured with a range of dispersive probes, J. Pharm. Pharmacol. Suppl. (57) (2005) S90–S91. [7] A. Voelkel, Inverse gas chromatography in characterization of surface, Chemometr. Intell. Lab. 72 (2004) 205–207. [8] M.A. Mohammad, Chromatographic adhesion law to simplify surface energy calculation, J. Chromatogr. A 1318 (2013) 270–275. [9] J.N. Miller, J.C. Miller, Statistics and Chemometrics for Analytical Chemistry, fifth ed., Pearson Education Limited, Harlow, 2005. [10] T. Perl, B. Bödeker, M. Jünger, J. Nolte, W. Vautz, Alignment of retention time obtained from multicapillary column gas chromatography used for VOC analysis with ion mobility spectrometry, Anal. Bioanal. Chem. 397 (2010) 2385–2394. [11] A. Barcaru, A. Anroedh-Sampat, H.G. Janssen, G. Vivó-Truyols, Retention time prediction in temperature-programmed, comprehensive two-dimensional gas chromatography: modeling and error assessment, J. Chromatogr. A 1368 (2014) 190–198. [12] Y. Koh, K.K. Pasikanti, C.W. Yap, E.C. Chan, Comparative evaluation of software for retention time alignment of gas chromatography/time-of-flight mass spectrometry-based metabonomic data, J. Chromatogr. A 1217 (2010) 8308–8316. [13] J.R. Conder, C.L. Young, Physicochemical Measurement by Gas Chromatography, Wiley-Interscience Publication, Chichester, 1979. [14] L. Wu, Y. Chen, S.A. Caccamise, Q.X. Li, Difference equation model for isothermal gas chromatography expresses retention behavior of homologues of n-alkanes excluding the influence of holdup time, J. Chromatogr. A 1260 (2012) 215–223. [15] Z. Yao, L. Ge, X. Ji, J. Tang, M. Xia, Y. Xi, Surface properties studies of bivalve shell waste by the IGC technique: probing its significant potential application in the polymer industry, J. Alloy Compd. 621 (2015) 389–395.
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Short communication
Accuracy verification of surface energy components measured by inverse gas chromatography Mohammad Amin Mohammad a,b,∗ a b
Department of Pharmacy and Pharmacology, University of Bath, Bath BA2 7AY, UK Faculty of Pharmacy, University of Damascus, Damascus, Syria
a r t i c l e
i n f o
Article history: Received 21 January 2015 Received in revised form 22 April 2015 Accepted 22 April 2015 Available online 30 April 2015 Keywords: Accuracy verification criteria Adhesion retention factor Chromatographic adhesion law Inverse gas chromatography Surface components
a b s t r a c t Inverse gas chromatography (IGC) measures the retention times of probes which are then used to calculate the surface properties of solids. No method is available to verify how much the measured values are close to their accurate values. According to the chromatographic adhesion law, the accurate determination of a the dispersive retention factor (KCH ) is a necessary prerequisite to obtain accurate surface components. 2
Employing two equations in this paper, %Sd , which is the percentage deviation of dispersive component d (Sd ) from its accurate value (S0 ), was correlated firstly to %CVln K a , the percentage coefficient of variation CH2
a a of ln KCH , and secondly to FEKaCH2 , the factor error of KCH , via two linear equations. The first equation is to 2
2
d , and the second equation is to estimate outline the upper and lower limits of the uncertainty range of S0 d d S0 . To minimize the uncertainty range of S0 to less than ±5%, %CVln K a should be less than 0.7%. Then CH2
considering the sign of FEKaCH2 narrows the uncertainty range to either its upper or lower half. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Inverse gas chromatography (IGC) is a simple, fast and accurate technique for characterizing the surface properties of solids in any form [1]. IGC is used to evaluate the dispersive surface energy (Sd ) of the solids using either Dorris–Gray [2] or Schultz [3] calculation methods which are based on the net retention volumes (Vn ) of n-alkane probes at infinite dilution. The electron acceptor (S+ ) and the electron donor (S− ) components of the solids are then calculated by subtracting the dispersive component from the total free energy of adsorption using the van Oss–Good–Chaudhury method [4]. These methods have been recently reviewed in details [1]. To enhance the accuracy of the obtained values, various researchers recently scrutinized the calculation methods. Shi et al. found that the values of Sd vary according to the calculation methods, and then they normalized this variation by determining the ratio between Dorris–Gray and Schultz methods [5]. It was also found that the calculated surface components of the same sample vary up to 16% depending on the used n-alkane series, and they attributed this discrepancy to uneven differences in the literature parameters of n-alkanes which are their dispersive free energy (ld )
∗ Department of Pharmacy and Pharmacology, University of Bath, Bath BA2 7AY, UK. Tel.: +44 1225 386797. E-mail address: [email protected] http://dx.doi.org/10.1016/j.chroma.2015.04.044 0021-9673/© 2015 Elsevier B.V. All rights reserved.
and cross-sectional area (˛) [6]. The traditional literature parameters of n-alkanes listed in the papers using Schultz method are not accurate enough [7] and Shi et al. reported [5] that they are not in good agreement with the same parameters in solvents handbook. To overcome calculated IGC data variability, Mohammad combined the van Oss–Good–Chaudhury concept, the Dorris–Gray equation, the Schultz equation, the Fowkes equation and group contribution theory to establish the chromatographic adhesion law [8]. This law a ), the electron acceptor uses the dispersive retention factor (KCH 2 a ), and the electron donor retention factor (K a ) retention factor (Kl+ l− a and K a valto calculate and characterise the surface properties. Kl+ l− a , and they are calculated from the net retention ues depends on KCH 2 times of the probes (tn ) obtained from the measured retention times of the probes (tr ). Chromatographic apparatus has been designed in order to obtain measurements of high precision. However, random errors can be minimized but not eliminated [9]. The tolerances and the failures of the equipment used for the control of the temperature and flow rate, faulty operation, and sampling procedure cause variations of column temperature and carrier gas flow rate which, in turn, cause significant changes in tr s [10,11]. tr s also change with variation in column performance or column overloading with sample [12]. These potential variations in tr s of a probe or many probes a , and so the calculated surface properties will change tn , Vn and KCH 2 deviate from their accurate values.
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
Researchers have published several hundred papers on surface characterization with IGC. Their criterion to judge the accuracy of IGC measurements for tr s is the linearity of the n-alkane line “r”, 0.5
which is the plot of RTlnVn versus ˛(ld ) . Usually, “r” should be ≥0.9995 [13]. To date, no method has been reported to verify the accuracy of IGC measurement for tr s, and to evaluate how tr s’ inaccuracy affects the accuracy of obtained surface properties. In a these studies, the dependence of Sd , S− and S+ on KCH is clarified 2 a and then KCH is used to establish the accuracy verification criteria 2 of IGC measurements. These criteria will be used to evaluate the d ). deviation of obtained Sd from its accurate value (S0 2. Discussion 2.1. Mathematical derivation The chromatographic adhesion law can be used to calculate Sd , + S and S− , and its general equation is: Kia
E a /kT
=e
i
(1)
Kia
Eia
is the adhesion retention factor of a chemical group, where represents the increased adhesion energy due to this group, k is the Boltzmann constant, and T is the column temperature (in degrees Kelvin). CH2 participates in the chromatographic adhesion via its dispera is calculated from tn s of any two sive component only, and its KCH 2 successive n-alkanes, as follows: tnci+1 tnci
the retention time of its theoretical n-alkane reference (tnl−,ref ), as follows: a Kl− =
ln tnl−,ref =
a El− = 2al− (l− )
a ECH = 2
(3)
In this case, the general equation becomes as follows: E a
a KCH =e
CH2
/kT
(4)
2
a Kl+
of a monopolar electron acceptor probe (l+) (e.g., dichloromethane or chloroform) can be calculated from its net retention time (tnl+ ) and the retention time of its theoretical nalkane reference (tnl+,ref ), as follows: a Kl+ =
tnl+ tnl+,ref
(5)
and tnl+,ref can be calculated as follows:
a ln tnl+,ref = ln tnci + ln KCH
2
d ) ˛l+ (l+
0.5
d ) ˛ci+1 (l,ci+1
d ) − ˛ci (l,ci
0.5
0.5
d ) − ˛ci (l,ci
0.5
a El+ = 2al+ (l+ )
0.5
(s− )
0.5
Sd =
E a /kT
a Kl+ =e
l+
(s+ )
d ) − ˛ci (l,ci
0.5
0.5
(10)
(11)
(12)
l−
2
˛2CH CH2 a )2 0.477 (T ln Kl+
a2l+ l+
a )2 0.477 (T ln Kl−
(mJ m−2 )
(13)
a2l− l−
(mJ m−2 )
(14)
(mJ m−2 )
(15)
where the units of ˛ are Å2 and of are mJ m−2 in all equations. Rounding off 0.477 to the nearest tenth (to be 0.5) will cause 4.5% increase in calculated surface components, and rounding off 0.477 to two decimal digits (to be 0.48) will cause 0.6% increase in calculated surface components. Eqs. (6), (10), (13)–(15) clarify that all surface components of a . Therefore, solids (Sd , S− and S+ ) depend on the value of KCH 2 a the accuracy of the measured value of KCH is a prerequisite for 2
verifying the accuracy of the calculated values of Sd , S− and S+ . Supposedly, five n-alkanes (n-hexane to n-decane) are used to a can be calculated from the net probe a solid. In this case, KCH 2 retention times of any two successive n-alkanes, as follows: =
tn7 tn6
(16)
=
tn8 tn7
(17)
=
tn9 tn8
(18)
2
CHC8,C7 2
Ka
CHC9,C8 2
Ka
CHC10,C9 2
=
tn10 tn9
(19)
tn is calculated as follows: tn = (tr − t0 )
a of a monopolar electron donor probe (l−) (e.g., ethyl acetate Kl− or toluene) can be calculated from its net retention time (tnl− ) and
0.5
a )2 0.477 (T ln KCH
Ka
(8)
0.5
0.5
2
(6)
In this case, the general equation becomes as follows:
d ) ˛ci+1 (l,ci+1
d ) − ˛ci (l,ci
Combining Eqs. (3), (7) and (11) with Eqs. (4), (8) and (12), respectively, produces the following similar equations to calculate the three surface components of the solids:
CHC7,C6
(7)
0.5
E a /kT
a Kl− =e
Ka
d and a are the dispersive component of l+ and its where l+ l+ cross-sectional area. As the dispersive component of l+ is omitted by its theoretical n-alkane, l+ contributes to the chromatographic adhesion energy via its electron acceptor component (l+ ), and its a ) is calculated as follows: additive adhesion energy (El+
d ) ˛l− (l−
In this case, the general equation becomes as follows:
s+ =
0.5 2˛CH2 (CH2 )0.5 (Sd )
a ln tnci + ln KCH 2
d and a are the dispersive component of l− and its crosswhere l− l− sectional area. As the dispersive component of l− is omitted by its theoretical n-alkane, l− contributes to the chromatographic adhesion energy via its electron donor component (l− ) only, and its a ) is calculated as follows: additive adhesion energy (El−
The additive contribution of CH2 to the adhesion energy is only a ) and it is calculated as follows: dispersive (ECH 2
(9)
s− =
2
tnl− tnl−,ref
tnl−,ref can be calculated as follows:
(2)
a KCH =
89
(20)
where t0 is the retention time of a non-adsorbing marker. Combining the logarithms of any combination of Eqs. (16)–(19) a yields that ln KCH can be calculated from the subtraction of the 2 logarithms of the net retention times (lntn ) of any two n-alkanes
90
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
divided by the subtraction of their carbon atom numbers (n), which can be translated to the following equation: a ln KCH = 2
ln tn n
(21)
Integration of Eq. (21) gives: (22)
2
2.2. IGC experiment
The slope of Eq. (22), which is the same as that of Conder–Young a [14], averages out ln KCH from the n-alkane line (lntn vs. n). 2 Ideally, if trC6 , trC7 , trC8 , trC9 and trC10 are measured accurately, the standard deviation of the slope of Eq. (22) (SDln K a ) and its CH2
percentage coefficient of variation (%CVln K a
CH2
) are zero. Practically,
they have positive values, because there are experimental errors in tr s’ measurements as set out in the introduction. %CVln K a is
CH2
%CVln K a
CH2
SDln Ka
CH2
=
× 100
a ln KCH
(23)
CH2
increases, the possible deviation of Sd from its
d ) increases, and the percentage deviation of d accurate value (S0 S d from S0 is:
d ) (Sd − S0
× 100
d S0
(24)
CH2
a be used to determine the sign of %Sd . The factor error of KCH
2
(FEKaCH ), which has zero (ideally), positive or negative values, and 2
so it would be correlated with the sign of %Sd . FEKaCH was derived 2 from Eqs. (16) to (19) as follows, if tr s was determined accurately: C7,C6 2
CH
ln K a
= Ka
= Ka
C8,C7 2
CH
C7,C6 2
CH
(ln K a
C7,C6 CH 2
C9,C8 2
CH
= ln K a
C8,C7 2
CH
+ ln K a
C8,C7 CH 2
= Ka
= ln K a
C9,C8 2
CH
− ln K a
C9,C8 CH 2
= ln K a
− ln K a
C10,C9 CH 2
+ ln K a C8,C7 CH
− ln K a
2
CH
C9,C8 2
− ln K a
CH
C10,C9 2
FEKaCH and %Sd are zero. Therefore, the obtained Sd can be con-
)
Combining Eqs. (16)–(19) with Eq. (28) produces: = 100 × ln
In general FEKaCH 2
= 100 × ln
(tnCi )2 tnCi−2 tnCi+2
0.5
) × 100 = 0 (27)
(28)
(tnC8 )2 tnC6 tnC10
0.5
d d ) ) − ˛ci (l,ci of six n-alkanes (from n-pentane to of ˛ci+1 (l,ci+1 n-decane) listed in the Schultz and co-workers paper [3]. However, according to Dorris–Gray [2], ˛CH2 = 6 A˚ 2 and CH2 is calculated as follows:
The laboratory experiment was repeated until we obtained retention times of the five n-alkanes “r” = 1.00000, and %CVln K a ,
2
and (22), respectively. Then Sd was calculated using Eq. (13) of the chromatographic adhesion law. Also, Sd was calculated according to the Schultz and Dorris-Gray methods for reference. In Eq. (13), we set ˛CH2 (CH2 )0.5 equal to 35.72 A˚ 2 (mJ/m2 )0.5 , which is the average
(26)
C10,C9 2
CH
× 100 = FEKaCH
FEKaCH 2
where n is the carbon number of the homologous n-alkanes, and A and B are constants. This is because the linear relationship between ln(tr − t0 ) and n under isothermal conditions (Conder–Young concept) has been widely used (since the 1950s) to calculate t0 [14]. a using Eqs. (20) t0 was then used to calculate tn s, and ln KCH
CH2 = 35.6 + 0.058(293 − T ) (mJ m−2 )
2
2
(31)
(25)
C10,C9 2
CH
The left-hand side of Eq. (27) is zero in an ideal state. However, practically, the left-hand side can be zero, positive or negative, and considered as FEKaCH , so Eq. (27) becomes: (ln K a C7,C6 CH
ln(tr − t0 ) = An + B
2
%Sd can be zero (ideally), positive or negative, and the positive d towards higher values, and vice %Sd means Sd deviates from S0 versa. However, %CVln K a is either zero or positive, so it cannot
Ka
Experiments were performed using inverse gas chromatography (IGC 2000, Surface Measurement Systems Ltd., UK). Lactose monohydrate (Pharmatose 200M, Fonterra Ltd., New Zealand) (63–90 m) was packed into a pre-silanised glass column (300 mm × 3 mm i.d.). The column was analysed at 30 ◦ C and zero relative humidity, using anhydrous helium gas as the carrier. A series of n-alkanes (n-hexane to n-decane) was injected at the infinite dilution region. Their retention times followed from detection using a flame ionization detector (FID). t0 is considered to be the value that gives the best linearity in the Conder–Young equation:
2
When %CVln K a
%Sd =
2
%Sd .
a ln tn = ln KCH n + constant
calculated as follows:
Eq. (27) and therefore Eqs. (28)–(30) have six alternative forms as the four (equal) components within Eq. (27), when the total is set to = 0 requires two components to be added and two to be subtracted. The simulated experiments (Section 2.2 discussed below) show that FEKaCH , calculated using Eq. (30), is well correlated with
(29)
(30)
(32)
CH2
2
d which was 47.0 mJ m−2 as calculated by both Eq. (13) sidered as S0 of the chromatographic adhesion law and the Schultz method. Supposedly, one or more of tr s deviate during IGC measurements due to practical reasons or random errors. Therefore, tr s of the five n-alkanes would be shift up (+), down (−) or stay accurate (0). In this case, the number of possibilities is equal to 35 = 243; these possibilities are given codes starting from 1 until 243 (Supplementary data, Table 1S). Code 1 represents the original accurate tr s. Code 2 and code 3 represent another accurate sets of numbers, as tr s of all n-alkanes were shifted down and up simultaneously at the same rate (i.e., changing the flow rate), respectively. Thered . fore, simulated experiments with codes 1–3 generate the same S0 For other possibilities, we created 50 simulated experiments (50 sets of tr s) spreading over the traditional acceptable range of “r” [1–0.9995]. However, each set of tr s that produced negative t0 was omitted. The total number of experiments generated was 11,385 simulated experiments. For each experiment, Sd (using chromatoa , graphic adhesion law, Schultz, and Dorris-Gray methods), KCH 2
d , Again, if tr s of all n-alkanes is determined accurately, Sd is S0 a d and %CVln K a , FEKCH and %S are zero, also the linear fit degree CH2
2
for the plot of RTlnVn versus ˛(Ld )
0.5
becomes exactly one.
%CVln K a
CH2
, FEKaCH , %Sd , and “r” were calculated (Supplementary 2
data, Table 2S). Fig. 1 shows that Sd s calculated by both chromatographic adhesion law and Schultz method are similar and lower than those
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
91
0.5
Fig. 1. The possibilities of the dispersive surface energy values (Sd ) of lactose monohydrate as function of the linear fit degree “r” for the plot of RTlnVn versus ˛(Ld ) ; the dispersive surface energy values calculated by chromatographic adhesion law (blue), Schultz method (red) and Dorris-Gray method (green) (For interpretation of the color information in this figure legend, the reader is referred to the web version of the article.).
Fig. 2. The possibilities of the dispersive surface energy values (Sd ) of lactose monohydrate as function of the percentage coefficient of variation of dispersive retention factor (%CVln K a ); the dispersive surface energy values calculated by chromatographic adhesion law (blue), Schultz method (red) and Dorris-Gray method (green) (For CH2
interpretation of the color information in this figure legend, the reader is referred to the web version of the article.).
calculated by the Dorris–Gray method, and this observation is consistent with previous findings [5,8,15]. Also it shows that, irrespective of calculation methods, (Sd × 100)/(100 + 7.5%CVln K a ) CH2
d . When “r” approaches one, d varies by about ±15% from S0 S d . However, the possibilities of d values spread approaches S0 S widely (to lower and higher values) as “r” decreases from 1 toward 0.9995. Therefore, r ≥ 0.9995 is not a suitable criterion to accept d is IGC measurements because the uncertainty about Sd to be S0 high even at “r” close to one. However, Fig. 2 shows that %CVln K a
%CVln K a
CH2
(Fig. 3). Each value of %CVln K a
CH2
%Sd
is equivalent to a range
values spreading symmetrically about its zero value, the of distance between the upper and lower limits of the uncertainty (%upper Sd and %lower Sd ) is the uncertainty range about Sd to be d , this uncertainty range is outlined by two straight lines in Fig. 3 S0 which shows that %CVln K a linearly correlates with %upper Sd and CH2
%lower Sd with the following equation:
CH2
is a better criterion for the accuracy verification of IGC experid . The upper ments. At %CVln K a being zero or close to zero, Sd is S0 CH2
d increase and lower limits of the uncertainty about Sd to be S0 d linearly as functions of %CVln K a . As %S represents the percent-
age deviation of
Sd
CH2
from
d , S0
%Sd
was plotted as a function of
%limits Sd = ±7.5%CVln K a
(33)
CH2
The coefficient of determination “r2 of Eq. (33) is 1.000. This d of an investigated solid is with the means that we know that S0
92
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
d Fig. 3. The correlation between the limits of the percentage deviation of dispersive surface energy from its accurate value (the uncertainty range of S0 ) and the percentage coefficient of variation of dispersive retention factor (%CVln K a ). CH2
uncertainty range from (Sd × 100)/(100 + 7.5%CVln K a 100)/(100 − 7.5%CVln K E
CH2
) to (Sd ×
), and its equation can be represented as:
variables, all points were linearly regressed within the following linear equation:
CH2
%Sd = 0.3FEKaCH
(35)
2
d Uncertainty range of S0 = [(Sd × 100)/(100 + 7.5%CVln K a
CH2
(Sd × 100)/(100 − 7.5%CVln K a
CH2
) to
The correlation coefficient “r” of Eq. (35) is 0.88. This means that we can estimate that:
)] (34) d S0 =
(Sd × 100)
(36)
(100 + 0.35FEKaCH ) 2
If %CVln K a
CH2
d . The uncertainty equals zero, we know that Sd is S0
d increases as %CV range about S0 ln K a
CH2
increases, and vice versa.
d within We developed a mathematical model to estimate S0 its uncertainty range. FEKaCH was the best model we found. Fig. 4
Moreover, the sign of FEKaCH is the same as that of %Sd . This 2 helps to narrow the uncertainty range to either its upper or lower half, i.e., if FEKaCH is positive, Eq. (34) becomes: 2
2
shows the strong and direct correlation between FEKaCH and %Sd . 2 Most of the points form straight lines, but a few points form curves. Therefore, to approximate the correlation between these two
d Uncertainty range of S0
=
(Sd × 100)
(100 + 7.5%CVln K a
CH2
to (Sd ) )
(37)
Fig. 4. The correlation between the percentage deviation of dispersive surface energy from its accurate value (%Sd ) and the factor error of dispersive retention factor (FEKaCH ). 2
M.A. Mohammad / J. Chromatogr. A 1399 (2015) 88–93
93
Table 1 d ) and its estimated value for three different powders; A is lactose monohydrate, B is micronized lactose, The uncertainty range of the accurate dispersive surface energy (S0 and C is recrystallized lactose. Solid
trC6 (min)
trC7 (min)
trC8 (min)
trC9 (min)
trC10 (min)
%CVln K a
FEKaCH
Sd (mJ m−2 )
Uncertainty range d (mJ m−2 ) of S0
Narrowed uncertainty range of d (mJ m−2 ) by considering S0 the sign of FEKaCH
d Estimated S0 (mJ m−2 )
A B C A*
0.561 0.660 2.716 0.602
1.130 1.271 3.178 1.168
3.128 3.324 4.336 3.045
9.104 10.135 7.562 9.102
28.673 33.102 16.357 28.392
0.908 0.354 0.679 0.689
14.5 4.8 1.5 −6.8
48.9 51.1 34.6 45.6
45.7–52.4 49.8–52.5 32.9–36.4 43.4–48.1
45.7–48.9 49.8–51.1 32.9–34.6 45.6–48.1
46.8 50.4 34.4 46.6
CH2
2
2
*
The experiment of material A was repeated to get lower %CVln K a
CH2
.
and if FEKaCH is negative, Eq. (34) becomes: 2
d = Uncertainty range of S0
(Sd ) to
(Sd × 100)
(100 − 7.5%CVln K a )
(38)
CH2
%CVln K a
CH2
Acknowledgments
, FEKaCH , and %Sd are ratios (unit less) and 2
a , d , d or/and t s vary due to varying solids, unchangeable if KCH r S0 S 2 temperature, flow rate, calculation methods or/and experimental errors. Therefore, the above simulated experiments (∼12,000) on one solid are enough to verify our suggested criteria of the accuracy of IGC measurements. Table 1 shows some examples of data and the application of the d different from new criteria. The data show that powder C has S0 those of both powders A and B because its uncertainty range does not overlap with others’ uncertainty ranges. However, the uncertainty ranges of A and B overlap partially. Therefore, we repeated the IGC experiment for A to minimize the uncertainty range by d valdecreasing %CVln K a . As a result, the difference between S0 CH2
ues of A and B becomes clear as concluded from their separated uncertainty ranges (last row of Table 1). Also, the narrowed uncertainty ranges obtained by considering the sign of FEKaCH show the 2 differences between the three solids, even before retesting solid d A. Moreover, the values of estimated S0 quantify the difference between the solids’ dispersive surface components. All estimated d using Eq. (36) values are within the narrowed uncertainty S0 ranges calculated using Eqs. (37) and (38). This elucidates the harmony between Eqs. (36) and (34) and its narrowed forms, Eqs. (37) and (38). Three or four n-alkanes can be used to outline the uncertainty d as to calculate range, but they are not enough to estimate S0 a FEKCH five n-alkanes are required. Also, six n-alkanes are more 2 than enough. 3. Conclusions d , but it is possible to be Practically, it is difficult to measure S0 close to its accurate value. It is essential to have criteria to quantify the uncertainty about Sd . %CVln Ka has been used to outline the CH2
d . This uncertainty range uncertainty range, and FEKaCH estimates S0 2 can be narrowed by repeating the experiment to obtain lower values of %CVln Ka . We recommend IGC experiments with %CVln K a CH2
CH2
less than 0.7% to minimize the uncertainty about measured Sd to less than ±5%. For example, when %CVln K a is 0.7% and Sd is CH2
50 mJ m−2 , we know that the accurate value of Sd is within the range of 47.5–52.8 mJ m−2 . Lowering %CVln K a below 0.7% minimizes CH2
factor is a necessary prerequisite to obtain accurate surface components. The new criteria have been applied herein to verify the accuracy of the dispersive component.
this uncertainty range. Moreover, considering the sign of FEKaCH 2 narrows the uncertainty range by half. The method reported herein highlights that the accuracy of the measured dispersive retention
I gratefully acknowledge University of Bath and CARA (Zeid AlBayaty and Kate Robertson) for providing an academic fellowship, and thank Dr. Ian S. Blagbrough (University of Bath), Prof James H. Davenport (University of Bath), and Prof Victor Rivero (visiting professor, University of Bath and Jalisco Institution, Mexico) for their helpful discussions. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.chroma.2015.04. 044 References [1] S. Mohammadi-Jam, K.E. Waters, Inverse gas chromatography applications: a review, Adv. Colloid Interface Sci. 212 (2014) 21–44. [2] G.M. Dorris, D.G. Gray, Adsorption of n-alkanes at zero surface coverage on cellulose paper and wood fibers, J. Colloid Interface Sci. 77 (1980) 353–362. [3] J. Schultz, L. Lavielle, C. Martin, The role of the interface in carbon fibre-epoxy composites, J. Adhesion 23 (1987) 45–60. [4] C. van Oss, R. Good, M. Chaudhury, Additive and nonadditive surface tension components and the interpretation of contact angles, Langmuir 4 (1988) 884–891. [5] B. Shi, Y. Wang, L. Jia, Comparison of Dorris-Gray and Schultz methods for the calculation of surface dispersive free energy by inverse gas chromatography, J. Chromatogr. A 1218 (2011) 860–862. [6] M.A. Mohammad, I.M. Grimsey, R.T. Forbes, An approach to normalise inverse gas chromatography data measured with a range of dispersive probes, J. Pharm. Pharmacol. Suppl. (57) (2005) S90–S91. [7] A. Voelkel, Inverse gas chromatography in characterization of surface, Chemometr. Intell. Lab. 72 (2004) 205–207. [8] M.A. Mohammad, Chromatographic adhesion law to simplify surface energy calculation, J. Chromatogr. A 1318 (2013) 270–275. [9] J.N. Miller, J.C. Miller, Statistics and Chemometrics for Analytical Chemistry, fifth ed., Pearson Education Limited, Harlow, 2005. [10] T. Perl, B. Bödeker, M. Jünger, J. Nolte, W. Vautz, Alignment of retention time obtained from multicapillary column gas chromatography used for VOC analysis with ion mobility spectrometry, Anal. Bioanal. Chem. 397 (2010) 2385–2394. [11] A. Barcaru, A. Anroedh-Sampat, H.G. Janssen, G. Vivó-Truyols, Retention time prediction in temperature-programmed, comprehensive two-dimensional gas chromatography: modeling and error assessment, J. Chromatogr. A 1368 (2014) 190–198. [12] Y. Koh, K.K. Pasikanti, C.W. Yap, E.C. Chan, Comparative evaluation of software for retention time alignment of gas chromatography/time-of-flight mass spectrometry-based metabonomic data, J. Chromatogr. A 1217 (2010) 8308–8316. [13] J.R. Conder, C.L. Young, Physicochemical Measurement by Gas Chromatography, Wiley-Interscience Publication, Chichester, 1979. [14] L. Wu, Y. Chen, S.A. Caccamise, Q.X. Li, Difference equation model for isothermal gas chromatography expresses retention behavior of homologues of n-alkanes excluding the influence of holdup time, J. Chromatogr. A 1260 (2012) 215–223. [15] Z. Yao, L. Ge, X. Ji, J. Tang, M. Xia, Y. Xi, Surface properties studies of bivalve shell waste by the IGC technique: probing its significant potential application in the polymer industry, J. Alloy Compd. 621 (2015) 389–395.
