Journal of Chromatography A, 1408 (2015) 267–271
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Short communication
An equation to calculate the actual methylene middle parameter as a function of temperature 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 22 May 2015 Received in revised form 29 June 2015 Accepted 1 July 2015 Available online 3 July 2015 Keywords: Adhesion retention factor Chromatographic adhesion law Inverse gas chromatography Methylene middle parameter Surface energy Surface parameters of methylene
a b s t r a c t
, the product of the methylene group’s cross-sectional area
of the surface parameters of methylene (˛CH2 and CH2 ), and so
The surface energy of solids consists of dispersive (sd ) and specific (electron acceptor and electron donor) components. Inverse gas chromatography (IGC) is a common technique to measure these components. The accuracy of sd calculation influences the calculated values of both sd and the specific components [1]. The calculation of sd was simplified by combining the Dorris–Gray equation [2] and the Schultz equation [3] within the chromatographic adhesion law [4], and its simplified equation to calculate sd is: a )2 0.477(T ln KCH 2
˛2CH CH2
(˛CH2 ) and the root square of its dispersive free energy (CH2 ), is the key parameter to calculate the dispersive surface components of solids (sd ) using inverse gas chromatography (IGC) at different temper0.5 atures. The only method reported to calculate ˛CH2 (CH2 ) as a function of temperature is the Dorris–Gray 0.5 method. However, the conventional values of ˛CH2 (CH2 ) calculated by the Dorris–Gray method depend 0.5 heavily on theoretical aspects. This paper establishes a novel equation calculating the actual ˛CH2 (CH2 ) as a function of temperature using the latest and most accurate surface parameters of seven succes0.5 sive n-alkanes. The obtained actual ˛CH2 (CH2 ) values are slightly higher those of the conventional 0.5 0.5 ˛CH2 (CH2 ) . At 20 ◦ C, the actual ˛CH2 (CH2 ) generates sd values less than those generated using the 0.5 conventional ˛CH2 (CH2 ) by ∼3%, and this reduction in calculated sd values increases linearly to become 0.5 ◦ ∼5% at 100 C. Therefore, using the new actual ˛CH2 (CH2 ) seems to mitigate the discrepancy between the sd values measured by IGC and those measured by the contact angle method. © 2015 Elsevier B.V. All rights reserved.
1. Introduction
sd =
0.5
Methylene middle parameter ˛CH2 (CH2 )
mJ m−2
(1)
2
where ˛CH2 is the cross-sectional area of a methylene group in units of Å2 , CH2 is the dispersive free energy of the methylene group in a units of mJ m−2 , KCH is the dispersive retention factor, and T is the 2 column temperature (in degrees Kelvin). a The accuracy of KCH measurement and its effect on the accu2
racy of sd has been recently quantitated [5]. However, the values
E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.chroma.2015.07.003 0021-9673/© 2015 Elsevier B.V. All rights reserved.
the methylene middle parameter [˛CH2 (CH2 )0.5 ], are still contro-
versial. If ˛CH2 (CH2 )0.5 is calculated according to the Dorris–Gray method, Eq. (1) generates the same values as the Dorris–Gray method, and if ˛CH2 (CH2 )0.5 is calculated from the average of ˛Ci+1 (Cd
i+1
)
0.5
− ˛Ci (Cd )
0.5
i
, Eq. (1) generates the same values as the
Schultz method [4,5]. Materials are processed at different temperatures, and their performance during processing depends on their surface free energy. To date, Dorris–Gray [2] is the only reported method to calculate ˛CH2 (CH2 )0.5 as a function of temperature. They considered ˛CH2 as 6 A˚ 2 and its value is temperature independent, but they calculated CH2 as a function of temperature using the following equation: CH2 = 35.6 + 0.058(293 − T )mJ m−2
(2)
where T is the temperature (in degrees Kelvin). Then, 0.5 ˛CH2 (CH2 )
Dorris−Gray
is calculated using Eq. (2) and the
constant value of ˛CH2 being 6 A˚ 2 . Dorris and Gray calculated ˛CH2 by multiplying the separation distance of two carbon atoms of n-alkanes (which is 0.1275 nm) with the average distance between centers of CH2 groups in adjacent molecules (which is ≈0.47 nm). They derived Eq. (2) from
268
M.A. Mohammad / J. Chromatogr. A 1408 (2015) 267–271
the values of the surface tension of a linear polyethylene (containing only methylene groups) at different temperatures over the melt, and then from an extrapolation of the surface tenn-alkanes to infinite chain length. sion of low-molecular-weight Therefore, ˛CH2 (CH2 )0.5 depends heavily on theoretiDorris−Gray
cal due to the absence of an actual alternative of aspects. However, ˛CH2 (CH2 )0.5 , most researchers have used it to calcuDorris−Gray
late the dispersive surface energy at different temperatures [6–8]. In actuality, the methylene group represents the additive contribution of an n-alkane compared to its ascendant n-alkane to the adsorption to a tested solid. Therefore,
˛CH2 (CH2 )0.5
actual
= ˛Ci+1 (ld
Ci+1
)
0.5
− ˛Ci (ld )
0.5
(3)
Ci
2. Discussion Table 1 contains the latest data of n-alkanes’ surface tensions (ld ) of seven n-alkanes (n-pentane to n-undecane) within the temperature range 10–100 ◦ C taken from Handbook of Chemistry and Physics, Chemical Rubber Company [9]. This temperature range is appropriate to derive ˛CH2 (CH2 )0.5 because IGC is typically actual used to characterise the surface at fairly low temperatures (below 100 ◦ C) [10]. The last row of Table 1 contains the conventional values of ld which are usually taken from the latest SMS manual and from Schultz et al. [3]. The conventional ld values are available at 30 ◦ C only. Table 2 contains different ␣ values which are method dependent and temperature independent. These ˛ values were measured by the Kiselev method or calculated using theoretical calculations such as Van der Waals (VDW) model, Redlich–Kwong (R–K) equation, geometric model, cylindrical model or spherical model [11]. Different hypotheses were tested to study the effect of temperature on the values of ˛, and it was found that the hypothesis assuming that ˛ values are constant and temperature independent is the easiest method, which requires no additional assumptions, moreover, they fit the linearity of Eq. (4) of Schultz et al. [3] over a wide temperature range [10,12]. 0.5
˛(ld )
0.5
+C
(4)
where C is a constant, N is the Avogadro number, R is the gas constant, Vn indicates the net retention volumes of n-alkanes. These values of ld and ˛ listed in Tables 1 and 2 were used to
over the reported temperatures as calculate ˛CH2 (CH2 )0.5 actual follows: 0.5 ˛CH2 (CH2 )0.5 is the average of ˛Ci+1 (ld ) − actual
˛Ci (ld C
)
0.5
Ci +1
, and so it is calculated from:
i
˛CH2 (CH2 )0.5
actual
= ˛Ci+1 (ld
Ci +1
)
0.5
− ˛Ci (ld ) Ci
0.5
=
0.5 ˛(ld )
n (5)
where n is the carbon number of the homologous n-alkanes. Integration of Eq. (5) gives: 0.5 ˛(ld )
= ˛CH2 (CH2 )
0.5 actual
n + constant
lntn = An + B
(6)
(7)
where A and B are constants, tn indicates the net retention times of n-alkanes. Further details of the mathematical derivation of Eqs. (5) and (6) are reported in the supplementary information. The slope of Eq. (6) averages out ˛CH2 (CH2 )0.5 from the nalkane line of ˛(ld )
0.5
actual
versus n. The error% of the slope reflects the
accuracy of the obtained ˛CH2 (CH2 )0.5
actual
and the considered
n-alkanes’ surface parameters (ld and ˛). The error% is calculated from: Error% =
where ld and ␣ are the dispersive free energy and the crosssectional area of the homologous n-alkanes, respectively, and the subscripts (Ci+1 and Ci ) indicate any two successive n-alkanes. This paper aims to derive an equation calculating ˛CH2 (CH2 )0.5 actual as a function of temperature using the latest and most accurate n-alkane parameters.
RTln Vn = 2N(sd )
Also we can obtain Eqs. (5) and (6) by combining Eq. (4) of Schultz et al. [3] and Eq. (7) of Conder-Young [13].
Standard deviation of the slope Slope
× 100
(8)
when the error% approaches zero, the correlation coefficient of Eq. (6) approaches one, and the used data (˛ and ld ) obey Schultz and Conder-Young equations, the lower the error%, the more accu. rate ˛CH2 (CH2 )0.5 actual Table 3 shows the typical consistency between correlation coefficient “r” of Eq. (6) and the error% values of its slope. Table 3 also shows that when the ld values taken from the Handbook of Chemistry and Physics [9] of Kiselev results [11] are and the ˛ values reaches its lowest values used, the error% of ˛CH2 (CH2 )0.5 actual (less than 0.5%) at all temperatures with average of 0.2%. However, the average of error% increases when ˛ values of the other tabulated methods are used. Also the error% increases in the case of using the conventional ld (last row of Table 3). Although the conventional ld values (last row of Table 1) are still used by researchers, they lack accuracy. Shi et al. clarified the inaccuracy in the conventional ld values compared to those values reported in the solvents handbook [14]. This inaccuracy makes the calculated ld values depend on the n-alkane series used to
probe the solid [15]. Also, the conventional values of ld cover only one temperature (30 ◦ C). ˛CH2 calculated from ˛Ci+1 − ˛Ci of Kiselev results equals 6 A˚ 2 . However, other methods generate ˛CH2 either higher or lower than 6 A˚ 2 . Voelkel et al. cited different values of ˛CH2 ranging from 3.1 to 7.7 A˚ 2 , and they concluded that 6 A˚ 2 is the most accurate [16]. The above discussion and the values of “r” and Error% (Table 3) elucidate that the ld values listed in the Handbook of Chemistry and Physics and ˛ values of Kiselev results are the most accurate and convenient to calculate ˛CH2 (CH2 )0.5 actual compared to their counterpart values. Therefore, we used them as a functo derive the equation calculating ˛CH2 (CH2 )0.5
actual
tion of temperature. In this equation, we correlated ˛2CH CH2
2
actual
instead of ˛CH2 (CH2 )0.5 with temperature to obey the first actual order linear correlation between surface tensions of liquids and temperature, which is prominent in Eq. (2) and the data of Table 1. These data show that all n-alkanes’ ld values linearly correlate with temperature, and their Eqs. (9)–(15), which have correlation coefficient “r” 1.000000, are: d l,C5 = −0.111T + 48.486
(9)
d l,C6 = −0.102T + 48.379
(10)
d l,C7 = −0.099T + 49.068
(11)
d l,C8
= −0.095T + 49.494
(12)
d l,C9 = −0.094T + 50.268
(13)
d l,C10 = −0.092T + 50.801
(14)
d l,C11
(15)
= −0.090T + 51.084
M.A. Mohammad / J. Chromatogr. A 1408 (2015) 267–271
269
Table 1 The surface tension of n-alkanes (ld ) at different temperatures. ld (mJ m−2 )
Temperature (K)
a
283.16 298.16a 323.16a 348.16a 373.16a 303.16b a b
n-Pentane
n-Hexane
n-Heptane
n-Octane
n-Nonane
n-Decane
n-Undecane
17.15 15.49 n/a n/a n/a 16.00
19.42 17.89 15.33 n/a n/a 18.40
21.14 19.66 17.19 14.73 n/a 20.30
22.57 21.14 18.77 16.39 14.01 21.30
23.79 22.38 20.05 17.71 15.37 22.70
24.75 23.37 21.07 18.77 16.47 23.40
25.56 24.21 21.96 19.70 17.45 24.60
Data are taken from Ref. [9]. Data are taken from Ref. [3].
Table 2 The cross-sectional area of n-alkanes (˛) calculated using different methods. Method
˛ (Å2 ) n-Pentane
n-Hexane
n-Heptane
n-Octane
n-Nonane
n-Decane
n-Undecane
Kiselev results Cylindrical Geometric Spherical VDW R–K
46.0 39.3 32.9 36.4 47 36.8
51.5 45.5 40.7 39.6 52.8 41.3
57.3 51.8 48.5 42.7 59.2 46.4
63.0 58.1 56.3 45.7 64.9 50.8
69.0 64.4 64 48.7 69.6 54.5
75.0 70.7 71.8 51.7 74.4 58.2
81.0 n/a n/a n/a n/a n/a
Data are taken from Ref. [11].
Fig. 1 shows a first order linear correlation between temper, i.e., the actual CH2 , calculated in this ature and ˛2CH CH2 actual
2
paper from the most accurate n-alkane surface parameters, with correlation coefficient “r” is −0.999, described in Eq. (16):
˛2CH CH2 2
actual
4
= −1.869T + 1867.194 Å mJ m−2
(16)
Also, combining the value of ˛CH2 considered by the Dorris-Gray method within Eq. (2) gives:
˛2CH CH2 2
4
Dorris−Gray
= −2.088T + 1893.384 Å mJ m−2
(17)
Eq. (16) is very similar to Eq. (17). However, the subtraction Eq. (17) from Eq. (16) gives:
˛2CH CH2 2
actual
− ˛CH2 (CH2 )0.5
Dorris−Gray
4
= 0.219(T − 119.6) Å mJ m−2
(18)
To date,
˛2CH CH2
2
Dorris−Gray
has been used in calculation
sd
measured using IGC, which yielded higher of the values of values of sd compared to those measured using the contact angle method [17,18], and the discrepancy increases when temperature increases [10]. The reason for this discrepancy is that IGC is usually used at infinite dilution conditions at which n-alkanes preferentially probe the higher energy sites on the solid surface [19,20]. To probe the whole surface, Thielmann et al. developed the method of IGC at finite concentration conditions [21]. Then, Ho et al. compared the values of sd measured by IGC at finite concentration conditions with those measured by the contact angle method, and they found that IGC also generated higher values in the case of heterogeneous solid surfaces compared to the contact angle method, but similar values in the case of homogenous solid surfaces [22]. However, their comparisons were between the values measured using IGC at 30 ◦ C and the values measured by the contact angle method at 20 ◦ C. The inverse correlation between temperature and surface energy of solids is well [7]. Therefore, sd values mea established are always higher than sured by IGC using ˛2CH CH2 Dorris−Gray
2
those measured by the contact angle method. Also Shi et al. used the conventional ˛ and ld values of five n-alkanes (from n-hexane
to n-decane) to clarify the discrepancies in sd values generated using Schultz and Dorris–Gray methods at 30, 40 and 50 ◦ C, and d d they found that the ratio of s,Dorris-Gray /s,Schultz equals 1.02, 1.04, and 1.06, respectively [14]. Eq. (18) demonstrates that the values of ˛2CH CH2 are higher than those of
˛2CH CH2 2
Dorris−Gray
2
actual
at temperatures
d higher than (120 K, i.e., −153 ◦ C). This means that s,actual
calculated using ˛2CH CH2
2
culated using ˛2CH CH2
actual
d is lower than s,Dorris−Gray cal-
. For example, at temperature
Dorris−Gray (293.16 K), ˛2CH CH2 equals 1319.28 A˚ 4 mJ m−2 while 2 actual ˛2CH CH2 equals 1281.27 A˚ 4 mJ m−2 , and so Eq. (1) 2 Dorris−Gray
20 ◦ C
2
gives:
Fig. 1. The linear correlation between temperature and ˛2CH CH2 2
actual
calculated
using the latest ld values listed in the Handbook of Chemistry and Physics and ˛ values of Kiselev results.
˛2CH CH2
2
˛2CH CH2 2
Dorris−Gray actual
=
d s,actual d s,Dorris−Gray
= 0.97
(19)
R–K equation
27.58(2.8%) [0.998440] 27.42(2.9%) [0.998345] 26.25(3.6%) [0.998111] 24.59(3.2%) [0.998981] 23.03(1.2%) [0.999923]
2.7%
R–K equation
26.99(3.4%) [0.997757]
VDW equation
2.6%
VDW equation
34.50(3.2%) [0.997906]
M.A. Mohammad / J. Chromatogr. A 1408 (2015) 267–271
35.26(2.7%) [0.998549] 35.06(2.8%) [0.998451] 33.58(3.4%) [0.998279] 31.55(3.2%) [0.998957] 29.51(0.8%) [0.999964]
270
d d Fig. 2. The values of s,actual /s,Dorris−Gray as a function of the temperature between
versus n. 0.5 actual
20.78(2.5%) [0.998738] 43.17(0.4%) [0.999963] 36.96(0.7%) [0.999896] 303.16
35.99(0.8%) [0.999852]
Values within parenthesis are the error% of ˛CH2 (CH2 )
Kiselev results
0.5
. Values within square brackets are the correlation coefficient “r” of ˛(ld )
Spherical model Geometric model Cylindrical model
ld taken from Ref. [3], ␣ taken from Ref. [11] Temperature (K)
1.6% 0.3% 0.2% Average of Error%
0.3%
44.22(0.4%) [0.999966] 43.54(0.4%) [0.999972] 42.59(0.3%) [0.999986] 41.62(0.1%) [0.999998] 40.33(0.2%) [0.999998] 37.82(0.1%) [0.999999] 37.42(0.1%) [0.999998] 36.64(0.2%) [0.999994] 35.83(0.5%) [0.999975] 34.73(0.5%) [0.999989] 36.52(0.2%) [0.999994] 36.23(0.2%) [0.999994] 35.59(0.2%) [0.999995] 34.91(0.3%) [0.999988] 34.15(0.4%) [0.999985]
Spherical model Geometric model Cylindrical model Kiselev results
283.16 298.16 323.16 348.16 373.16
ld taken from Ref. [9], ˛ taken from Ref. [11]
versus n] Å2 (mJ m−2 )0.5 0.5
= Slope of [˛(ld ) actual
Temperature (K)
˛CH2 (CH2 )
0.5
Table 3 0.5 0.5 Methylene middle parameter ˛CH2 (CH2 ) calculated from n-alkane middle parameter ˛(ld ) at different temperatures.
21.20(1.6%) [0.999460] 21.22(1.9%) [0.999293] 20.56(1.5%) [0.999657] 20.02(1.7%) [0.999726] 19.38(1.5%) [0.999893]
293.16 and 373.16 K.
According to Eq. (18), the ratio of Eq. (19) linearly decreases further when temperature increases, e.g., it becomes 0.95 ◦ at 2 100 C (373.16 K) (Fig. 2). We can conclude that using ˛CH CH2 calculated using Eq. (16) instead of the conven2
tional
actual
˛2CH CH2 2
Dorris−Gray
seems to mitigate the discrepancy
sd
values measured by IGC and those measured by the between contact angle method, and also between sd values calculated using Schultz method and those calculated using Dorris-Gray method. a measured by IGC, Eq. (1) of chromatographic adheNow, KCH 2
sion law and Eq. (16) of ˛2CH CH2 2
actual
can be easily used to
calculate the dispersive surface energy at different temperatures almost without theoretical aspects. 3. Conclusions The literature surface parameters (˛ and ld ) of seven successive n-alkanes (n-pentane to n-undecane) were tested and it was found that the cross-sectional area of Kiselev results and the latest surface tensions (taken from the Handbook of Chemistry and Physics) are the most appropriate values to calculate the most accurate ˛CH2 (CH2 )0.5 with error% less than 0.5% at all temactual
peratures (from 10 to 100 ◦ C). The values of ˛2CH CH2 2
actual
were
then correlated with temperature to derive the equation calculating the actual methylene middle parameter as a function of temperature. The actual methylene middle parameter generates values of dispersive energy slightly lower than those generated by the conventional methylene middle parameter (by 3% at 30 ◦ C). Therefore, the values of the surface energy measured using IGC seems to become more similar to those values measured using the contact angle method if our new Eq. (16) is used to calculate the methylene middle parameter. Acknowledgments I gratefully acknowledge the University of Bath and CARA for providing an academic fellowship, and I thank Dr. Ian S. Blagbrough (University of Bath) for 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.2015.07. 003
M.A. Mohammad / J. Chromatogr. A 1408 (2015) 267–271
References [1] M.A. Mohammad, The Effect of the Molecular Conformation on the Surface Properties and Stability of Protein Particles, University of Bradford, Bradford, UK, 2006, pp. 97–100, PhD Thesis. [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] M.A. Mohammad, Chromatographic adhesion law to simplify surface energy calculation, J. Chromatogr. A 1318 (2013) 270–275. [5] M.A. Mohammad, Accuracy verification of surface energy components measured by inverse gas chromatography, J. Chromatogr. A 1399 (2015) 88–93. ˜ [6] M.R. Cuervo, E. Asedegbega-Nieto, E. Díaz, A. Vega, S. Ordónez, E. CastillejosLópez, I. Rodríguez-Ramos, Effect of carbon nanofiber functionalization on the adsorption properties of volatile organic compounds, J. Chromatogr. A 1188 (2008) 264–273. [7] H. Grajek, J. Paciura-Zadrozna, Z. Witkiewicz, Chromatographic characterisation of ordered mesoporous silicas: Part I. Surface free energy of adsorption, J. Chromatogr. A 1217 (2010) 3105–3115. [8] L. Qian, X. Lv, Y. Ren, H. Wang, G. Chen, Y. Wang, J. Shen, Inverse gas chromatography applied in the surface properties evaluation of mesocellular silica foams modified by sized nickel nanoparticles, J. Chromatogr. A 1322 (2013) 81–89. [9] W.M. Haynes, CRC Handbook of Chemistry and Physics, 95th ed., Taylor and Francis Group, LLC. Chemical Rubber Company (2014–2015), Surface Tension of Common Liquids, pp. 6-182 to 6-185, 2014–2015. [10] G. Garnier, W.G. Glasser, Measurement of the surface free energy of amorphous cellulose by alkane adsorption: a critical evaluation of Inverse Gas Chromatography (IGC), J. Adhes. 46 (1994) 165–180. [11] T. Hamieh, J. Schultz, New approach to characterise physicochemical properties of solid substrates by inverse gas chromatography at infinite dilution: I. some new methods to determine the surface areas of some molecules adsorbed on solid surfaces, J. Chromatogr. A 969 (2002) 17–25.
271
[12] A. Voelkel, Inverse gas chromatography in characterization of surface, Chemometr. Intell. Lab. 72 (2004) 205–207. [13] J.R. Conder, C.L. Young, Physicochemical Measurement by Gas Chromatography, Wiley-Interscience Publication, Chichester, 1979. [14] 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. [15] 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. 57 (Suppl.) (2005) S90–S91. [16] A. Voelkel, B. Strzemiecka, K. Adamska, K. Milczewska, Inverse gas chromatography as a source of physiochemical data, J. Chromatogr. A 1216 (2009) 1551–1566. [17] L. Segeren, M. Wouters, M. Bos, J. van den Berg, G. Vancso, Surface energy characteristics of toner particles by automated inverse gas chromatography, J. Chromatogr. A 969 (2002) 215–227. [18] S.K. Papadopoulou, G. Dritsas, I. Karapanagiotis, I. Zuburtikudis, C. Panayiotou, Surface characterization of poly(2,2,3,3,3-pentafluoropropyl methacrylate) by inverse gas chromatography and contact angle measurements, Eur. Polym. J. 46 (2010) 202–208. [19] N. Ahfat, G. Buckton, R. Burrows, M. Ticehurst, An exploration of interrelationships between contact angle, inverse phase gas chromatography and triboelectric charging data, Eur. J. Pharm. Sci. 9 (2000) 271–276. [20] H.E. Newell, G. Buckton, Inverse gas chromatography: investigating whether the technique preferentially probes high energy sites for mixtures of crystalline and amorphous lactose, Pharm. Res. 21 (2004) 1440–1444. [21] F. Thielmann, D.J. Burnett, J.Y. Heng, Determination of the surface energy distributions of different processed lactose, Drug Dev. Ind. Pharm. 33 (2007) 1240–1253. [22] R. Ho, S.J. Hinder, J.F. Watts, S.E. Dilworth, D.R. Williams, J.Y. Heng, Determination of surface heterogeneity of D-mannitol by sessile drop contact angle and finite concentration inverse gas chromatography, Int. J. Pharm. 387 (2010) 79–86.
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Short communication
An equation to calculate the actual methylene middle parameter as a function of temperature 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 22 May 2015 Received in revised form 29 June 2015 Accepted 1 July 2015 Available online 3 July 2015 Keywords: Adhesion retention factor Chromatographic adhesion law Inverse gas chromatography Methylene middle parameter Surface energy Surface parameters of methylene
a b s t r a c t
, the product of the methylene group’s cross-sectional area
of the surface parameters of methylene (˛CH2 and CH2 ), and so
The surface energy of solids consists of dispersive (sd ) and specific (electron acceptor and electron donor) components. Inverse gas chromatography (IGC) is a common technique to measure these components. The accuracy of sd calculation influences the calculated values of both sd and the specific components [1]. The calculation of sd was simplified by combining the Dorris–Gray equation [2] and the Schultz equation [3] within the chromatographic adhesion law [4], and its simplified equation to calculate sd is: a )2 0.477(T ln KCH 2
˛2CH CH2
(˛CH2 ) and the root square of its dispersive free energy (CH2 ), is the key parameter to calculate the dispersive surface components of solids (sd ) using inverse gas chromatography (IGC) at different temper0.5 atures. The only method reported to calculate ˛CH2 (CH2 ) as a function of temperature is the Dorris–Gray 0.5 method. However, the conventional values of ˛CH2 (CH2 ) calculated by the Dorris–Gray method depend 0.5 heavily on theoretical aspects. This paper establishes a novel equation calculating the actual ˛CH2 (CH2 ) as a function of temperature using the latest and most accurate surface parameters of seven succes0.5 sive n-alkanes. The obtained actual ˛CH2 (CH2 ) values are slightly higher those of the conventional 0.5 0.5 ˛CH2 (CH2 ) . At 20 ◦ C, the actual ˛CH2 (CH2 ) generates sd values less than those generated using the 0.5 conventional ˛CH2 (CH2 ) by ∼3%, and this reduction in calculated sd values increases linearly to become 0.5 ◦ ∼5% at 100 C. Therefore, using the new actual ˛CH2 (CH2 ) seems to mitigate the discrepancy between the sd values measured by IGC and those measured by the contact angle method. © 2015 Elsevier B.V. All rights reserved.
1. Introduction
sd =
0.5
Methylene middle parameter ˛CH2 (CH2 )
mJ m−2
(1)
2
where ˛CH2 is the cross-sectional area of a methylene group in units of Å2 , CH2 is the dispersive free energy of the methylene group in a units of mJ m−2 , KCH is the dispersive retention factor, and T is the 2 column temperature (in degrees Kelvin). a The accuracy of KCH measurement and its effect on the accu2
racy of sd has been recently quantitated [5]. However, the values
E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.chroma.2015.07.003 0021-9673/© 2015 Elsevier B.V. All rights reserved.
the methylene middle parameter [˛CH2 (CH2 )0.5 ], are still contro-
versial. If ˛CH2 (CH2 )0.5 is calculated according to the Dorris–Gray method, Eq. (1) generates the same values as the Dorris–Gray method, and if ˛CH2 (CH2 )0.5 is calculated from the average of ˛Ci+1 (Cd
i+1
)
0.5
− ˛Ci (Cd )
0.5
i
, Eq. (1) generates the same values as the
Schultz method [4,5]. Materials are processed at different temperatures, and their performance during processing depends on their surface free energy. To date, Dorris–Gray [2] is the only reported method to calculate ˛CH2 (CH2 )0.5 as a function of temperature. They considered ˛CH2 as 6 A˚ 2 and its value is temperature independent, but they calculated CH2 as a function of temperature using the following equation: CH2 = 35.6 + 0.058(293 − T )mJ m−2
(2)
where T is the temperature (in degrees Kelvin). Then, 0.5 ˛CH2 (CH2 )
Dorris−Gray
is calculated using Eq. (2) and the
constant value of ˛CH2 being 6 A˚ 2 . Dorris and Gray calculated ˛CH2 by multiplying the separation distance of two carbon atoms of n-alkanes (which is 0.1275 nm) with the average distance between centers of CH2 groups in adjacent molecules (which is ≈0.47 nm). They derived Eq. (2) from
268
M.A. Mohammad / J. Chromatogr. A 1408 (2015) 267–271
the values of the surface tension of a linear polyethylene (containing only methylene groups) at different temperatures over the melt, and then from an extrapolation of the surface tenn-alkanes to infinite chain length. sion of low-molecular-weight Therefore, ˛CH2 (CH2 )0.5 depends heavily on theoretiDorris−Gray
cal due to the absence of an actual alternative of aspects. However, ˛CH2 (CH2 )0.5 , most researchers have used it to calcuDorris−Gray
late the dispersive surface energy at different temperatures [6–8]. In actuality, the methylene group represents the additive contribution of an n-alkane compared to its ascendant n-alkane to the adsorption to a tested solid. Therefore,
˛CH2 (CH2 )0.5
actual
= ˛Ci+1 (ld
Ci+1
)
0.5
− ˛Ci (ld )
0.5
(3)
Ci
2. Discussion Table 1 contains the latest data of n-alkanes’ surface tensions (ld ) of seven n-alkanes (n-pentane to n-undecane) within the temperature range 10–100 ◦ C taken from Handbook of Chemistry and Physics, Chemical Rubber Company [9]. This temperature range is appropriate to derive ˛CH2 (CH2 )0.5 because IGC is typically actual used to characterise the surface at fairly low temperatures (below 100 ◦ C) [10]. The last row of Table 1 contains the conventional values of ld which are usually taken from the latest SMS manual and from Schultz et al. [3]. The conventional ld values are available at 30 ◦ C only. Table 2 contains different ␣ values which are method dependent and temperature independent. These ˛ values were measured by the Kiselev method or calculated using theoretical calculations such as Van der Waals (VDW) model, Redlich–Kwong (R–K) equation, geometric model, cylindrical model or spherical model [11]. Different hypotheses were tested to study the effect of temperature on the values of ˛, and it was found that the hypothesis assuming that ˛ values are constant and temperature independent is the easiest method, which requires no additional assumptions, moreover, they fit the linearity of Eq. (4) of Schultz et al. [3] over a wide temperature range [10,12]. 0.5
˛(ld )
0.5
+C
(4)
where C is a constant, N is the Avogadro number, R is the gas constant, Vn indicates the net retention volumes of n-alkanes. These values of ld and ˛ listed in Tables 1 and 2 were used to
over the reported temperatures as calculate ˛CH2 (CH2 )0.5 actual follows: 0.5 ˛CH2 (CH2 )0.5 is the average of ˛Ci+1 (ld ) − actual
˛Ci (ld C
)
0.5
Ci +1
, and so it is calculated from:
i
˛CH2 (CH2 )0.5
actual
= ˛Ci+1 (ld
Ci +1
)
0.5
− ˛Ci (ld ) Ci
0.5
=
0.5 ˛(ld )
n (5)
where n is the carbon number of the homologous n-alkanes. Integration of Eq. (5) gives: 0.5 ˛(ld )
= ˛CH2 (CH2 )
0.5 actual
n + constant
lntn = An + B
(6)
(7)
where A and B are constants, tn indicates the net retention times of n-alkanes. Further details of the mathematical derivation of Eqs. (5) and (6) are reported in the supplementary information. The slope of Eq. (6) averages out ˛CH2 (CH2 )0.5 from the nalkane line of ˛(ld )
0.5
actual
versus n. The error% of the slope reflects the
accuracy of the obtained ˛CH2 (CH2 )0.5
actual
and the considered
n-alkanes’ surface parameters (ld and ˛). The error% is calculated from: Error% =
where ld and ␣ are the dispersive free energy and the crosssectional area of the homologous n-alkanes, respectively, and the subscripts (Ci+1 and Ci ) indicate any two successive n-alkanes. This paper aims to derive an equation calculating ˛CH2 (CH2 )0.5 actual as a function of temperature using the latest and most accurate n-alkane parameters.
RTln Vn = 2N(sd )
Also we can obtain Eqs. (5) and (6) by combining Eq. (4) of Schultz et al. [3] and Eq. (7) of Conder-Young [13].
Standard deviation of the slope Slope
× 100
(8)
when the error% approaches zero, the correlation coefficient of Eq. (6) approaches one, and the used data (˛ and ld ) obey Schultz and Conder-Young equations, the lower the error%, the more accu. rate ˛CH2 (CH2 )0.5 actual Table 3 shows the typical consistency between correlation coefficient “r” of Eq. (6) and the error% values of its slope. Table 3 also shows that when the ld values taken from the Handbook of Chemistry and Physics [9] of Kiselev results [11] are and the ˛ values reaches its lowest values used, the error% of ˛CH2 (CH2 )0.5 actual (less than 0.5%) at all temperatures with average of 0.2%. However, the average of error% increases when ˛ values of the other tabulated methods are used. Also the error% increases in the case of using the conventional ld (last row of Table 3). Although the conventional ld values (last row of Table 1) are still used by researchers, they lack accuracy. Shi et al. clarified the inaccuracy in the conventional ld values compared to those values reported in the solvents handbook [14]. This inaccuracy makes the calculated ld values depend on the n-alkane series used to
probe the solid [15]. Also, the conventional values of ld cover only one temperature (30 ◦ C). ˛CH2 calculated from ˛Ci+1 − ˛Ci of Kiselev results equals 6 A˚ 2 . However, other methods generate ˛CH2 either higher or lower than 6 A˚ 2 . Voelkel et al. cited different values of ˛CH2 ranging from 3.1 to 7.7 A˚ 2 , and they concluded that 6 A˚ 2 is the most accurate [16]. The above discussion and the values of “r” and Error% (Table 3) elucidate that the ld values listed in the Handbook of Chemistry and Physics and ˛ values of Kiselev results are the most accurate and convenient to calculate ˛CH2 (CH2 )0.5 actual compared to their counterpart values. Therefore, we used them as a functo derive the equation calculating ˛CH2 (CH2 )0.5
actual
tion of temperature. In this equation, we correlated ˛2CH CH2
2
actual
instead of ˛CH2 (CH2 )0.5 with temperature to obey the first actual order linear correlation between surface tensions of liquids and temperature, which is prominent in Eq. (2) and the data of Table 1. These data show that all n-alkanes’ ld values linearly correlate with temperature, and their Eqs. (9)–(15), which have correlation coefficient “r” 1.000000, are: d l,C5 = −0.111T + 48.486
(9)
d l,C6 = −0.102T + 48.379
(10)
d l,C7 = −0.099T + 49.068
(11)
d l,C8
= −0.095T + 49.494
(12)
d l,C9 = −0.094T + 50.268
(13)
d l,C10 = −0.092T + 50.801
(14)
d l,C11
(15)
= −0.090T + 51.084
M.A. Mohammad / J. Chromatogr. A 1408 (2015) 267–271
269
Table 1 The surface tension of n-alkanes (ld ) at different temperatures. ld (mJ m−2 )
Temperature (K)
a
283.16 298.16a 323.16a 348.16a 373.16a 303.16b a b
n-Pentane
n-Hexane
n-Heptane
n-Octane
n-Nonane
n-Decane
n-Undecane
17.15 15.49 n/a n/a n/a 16.00
19.42 17.89 15.33 n/a n/a 18.40
21.14 19.66 17.19 14.73 n/a 20.30
22.57 21.14 18.77 16.39 14.01 21.30
23.79 22.38 20.05 17.71 15.37 22.70
24.75 23.37 21.07 18.77 16.47 23.40
25.56 24.21 21.96 19.70 17.45 24.60
Data are taken from Ref. [9]. Data are taken from Ref. [3].
Table 2 The cross-sectional area of n-alkanes (˛) calculated using different methods. Method
˛ (Å2 ) n-Pentane
n-Hexane
n-Heptane
n-Octane
n-Nonane
n-Decane
n-Undecane
Kiselev results Cylindrical Geometric Spherical VDW R–K
46.0 39.3 32.9 36.4 47 36.8
51.5 45.5 40.7 39.6 52.8 41.3
57.3 51.8 48.5 42.7 59.2 46.4
63.0 58.1 56.3 45.7 64.9 50.8
69.0 64.4 64 48.7 69.6 54.5
75.0 70.7 71.8 51.7 74.4 58.2
81.0 n/a n/a n/a n/a n/a
Data are taken from Ref. [11].
Fig. 1 shows a first order linear correlation between temper, i.e., the actual CH2 , calculated in this ature and ˛2CH CH2 actual
2
paper from the most accurate n-alkane surface parameters, with correlation coefficient “r” is −0.999, described in Eq. (16):
˛2CH CH2 2
actual
4
= −1.869T + 1867.194 Å mJ m−2
(16)
Also, combining the value of ˛CH2 considered by the Dorris-Gray method within Eq. (2) gives:
˛2CH CH2 2
4
Dorris−Gray
= −2.088T + 1893.384 Å mJ m−2
(17)
Eq. (16) is very similar to Eq. (17). However, the subtraction Eq. (17) from Eq. (16) gives:
˛2CH CH2 2
actual
− ˛CH2 (CH2 )0.5
Dorris−Gray
4
= 0.219(T − 119.6) Å mJ m−2
(18)
To date,
˛2CH CH2
2
Dorris−Gray
has been used in calculation
sd
measured using IGC, which yielded higher of the values of values of sd compared to those measured using the contact angle method [17,18], and the discrepancy increases when temperature increases [10]. The reason for this discrepancy is that IGC is usually used at infinite dilution conditions at which n-alkanes preferentially probe the higher energy sites on the solid surface [19,20]. To probe the whole surface, Thielmann et al. developed the method of IGC at finite concentration conditions [21]. Then, Ho et al. compared the values of sd measured by IGC at finite concentration conditions with those measured by the contact angle method, and they found that IGC also generated higher values in the case of heterogeneous solid surfaces compared to the contact angle method, but similar values in the case of homogenous solid surfaces [22]. However, their comparisons were between the values measured using IGC at 30 ◦ C and the values measured by the contact angle method at 20 ◦ C. The inverse correlation between temperature and surface energy of solids is well [7]. Therefore, sd values mea established are always higher than sured by IGC using ˛2CH CH2 Dorris−Gray
2
those measured by the contact angle method. Also Shi et al. used the conventional ˛ and ld values of five n-alkanes (from n-hexane
to n-decane) to clarify the discrepancies in sd values generated using Schultz and Dorris–Gray methods at 30, 40 and 50 ◦ C, and d d they found that the ratio of s,Dorris-Gray /s,Schultz equals 1.02, 1.04, and 1.06, respectively [14]. Eq. (18) demonstrates that the values of ˛2CH CH2 are higher than those of
˛2CH CH2 2
Dorris−Gray
2
actual
at temperatures
d higher than (120 K, i.e., −153 ◦ C). This means that s,actual
calculated using ˛2CH CH2
2
culated using ˛2CH CH2
actual
d is lower than s,Dorris−Gray cal-
. For example, at temperature
Dorris−Gray (293.16 K), ˛2CH CH2 equals 1319.28 A˚ 4 mJ m−2 while 2 actual ˛2CH CH2 equals 1281.27 A˚ 4 mJ m−2 , and so Eq. (1) 2 Dorris−Gray
20 ◦ C
2
gives:
Fig. 1. The linear correlation between temperature and ˛2CH CH2 2
actual
calculated
using the latest ld values listed in the Handbook of Chemistry and Physics and ˛ values of Kiselev results.
˛2CH CH2
2
˛2CH CH2 2
Dorris−Gray actual
=
d s,actual d s,Dorris−Gray
= 0.97
(19)
R–K equation
27.58(2.8%) [0.998440] 27.42(2.9%) [0.998345] 26.25(3.6%) [0.998111] 24.59(3.2%) [0.998981] 23.03(1.2%) [0.999923]
2.7%
R–K equation
26.99(3.4%) [0.997757]
VDW equation
2.6%
VDW equation
34.50(3.2%) [0.997906]
M.A. Mohammad / J. Chromatogr. A 1408 (2015) 267–271
35.26(2.7%) [0.998549] 35.06(2.8%) [0.998451] 33.58(3.4%) [0.998279] 31.55(3.2%) [0.998957] 29.51(0.8%) [0.999964]
270
d d Fig. 2. The values of s,actual /s,Dorris−Gray as a function of the temperature between
versus n. 0.5 actual
20.78(2.5%) [0.998738] 43.17(0.4%) [0.999963] 36.96(0.7%) [0.999896] 303.16
35.99(0.8%) [0.999852]
Values within parenthesis are the error% of ˛CH2 (CH2 )
Kiselev results
0.5
. Values within square brackets are the correlation coefficient “r” of ˛(ld )
Spherical model Geometric model Cylindrical model
ld taken from Ref. [3], ␣ taken from Ref. [11] Temperature (K)
1.6% 0.3% 0.2% Average of Error%
0.3%
44.22(0.4%) [0.999966] 43.54(0.4%) [0.999972] 42.59(0.3%) [0.999986] 41.62(0.1%) [0.999998] 40.33(0.2%) [0.999998] 37.82(0.1%) [0.999999] 37.42(0.1%) [0.999998] 36.64(0.2%) [0.999994] 35.83(0.5%) [0.999975] 34.73(0.5%) [0.999989] 36.52(0.2%) [0.999994] 36.23(0.2%) [0.999994] 35.59(0.2%) [0.999995] 34.91(0.3%) [0.999988] 34.15(0.4%) [0.999985]
Spherical model Geometric model Cylindrical model Kiselev results
283.16 298.16 323.16 348.16 373.16
ld taken from Ref. [9], ˛ taken from Ref. [11]
versus n] Å2 (mJ m−2 )0.5 0.5
= Slope of [˛(ld ) actual
Temperature (K)
˛CH2 (CH2 )
0.5
Table 3 0.5 0.5 Methylene middle parameter ˛CH2 (CH2 ) calculated from n-alkane middle parameter ˛(ld ) at different temperatures.
21.20(1.6%) [0.999460] 21.22(1.9%) [0.999293] 20.56(1.5%) [0.999657] 20.02(1.7%) [0.999726] 19.38(1.5%) [0.999893]
293.16 and 373.16 K.
According to Eq. (18), the ratio of Eq. (19) linearly decreases further when temperature increases, e.g., it becomes 0.95 ◦ at 2 100 C (373.16 K) (Fig. 2). We can conclude that using ˛CH CH2 calculated using Eq. (16) instead of the conven2
tional
actual
˛2CH CH2 2
Dorris−Gray
seems to mitigate the discrepancy
sd
values measured by IGC and those measured by the between contact angle method, and also between sd values calculated using Schultz method and those calculated using Dorris-Gray method. a measured by IGC, Eq. (1) of chromatographic adheNow, KCH 2
sion law and Eq. (16) of ˛2CH CH2 2
actual
can be easily used to
calculate the dispersive surface energy at different temperatures almost without theoretical aspects. 3. Conclusions The literature surface parameters (˛ and ld ) of seven successive n-alkanes (n-pentane to n-undecane) were tested and it was found that the cross-sectional area of Kiselev results and the latest surface tensions (taken from the Handbook of Chemistry and Physics) are the most appropriate values to calculate the most accurate ˛CH2 (CH2 )0.5 with error% less than 0.5% at all temactual
peratures (from 10 to 100 ◦ C). The values of ˛2CH CH2 2
actual
were
then correlated with temperature to derive the equation calculating the actual methylene middle parameter as a function of temperature. The actual methylene middle parameter generates values of dispersive energy slightly lower than those generated by the conventional methylene middle parameter (by 3% at 30 ◦ C). Therefore, the values of the surface energy measured using IGC seems to become more similar to those values measured using the contact angle method if our new Eq. (16) is used to calculate the methylene middle parameter. Acknowledgments I gratefully acknowledge the University of Bath and CARA for providing an academic fellowship, and I thank Dr. Ian S. Blagbrough (University of Bath) for 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.2015.07. 003
M.A. Mohammad / J. Chromatogr. A 1408 (2015) 267–271
References [1] M.A. Mohammad, The Effect of the Molecular Conformation on the Surface Properties and Stability of Protein Particles, University of Bradford, Bradford, UK, 2006, pp. 97–100, PhD Thesis. [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] M.A. Mohammad, Chromatographic adhesion law to simplify surface energy calculation, J. Chromatogr. A 1318 (2013) 270–275. [5] M.A. Mohammad, Accuracy verification of surface energy components measured by inverse gas chromatography, J. Chromatogr. A 1399 (2015) 88–93. ˜ [6] M.R. Cuervo, E. Asedegbega-Nieto, E. Díaz, A. Vega, S. Ordónez, E. CastillejosLópez, I. Rodríguez-Ramos, Effect of carbon nanofiber functionalization on the adsorption properties of volatile organic compounds, J. Chromatogr. A 1188 (2008) 264–273. [7] H. Grajek, J. Paciura-Zadrozna, Z. Witkiewicz, Chromatographic characterisation of ordered mesoporous silicas: Part I. Surface free energy of adsorption, J. Chromatogr. A 1217 (2010) 3105–3115. [8] L. Qian, X. Lv, Y. Ren, H. Wang, G. Chen, Y. Wang, J. Shen, Inverse gas chromatography applied in the surface properties evaluation of mesocellular silica foams modified by sized nickel nanoparticles, J. Chromatogr. A 1322 (2013) 81–89. [9] W.M. Haynes, CRC Handbook of Chemistry and Physics, 95th ed., Taylor and Francis Group, LLC. Chemical Rubber Company (2014–2015), Surface Tension of Common Liquids, pp. 6-182 to 6-185, 2014–2015. [10] G. Garnier, W.G. Glasser, Measurement of the surface free energy of amorphous cellulose by alkane adsorption: a critical evaluation of Inverse Gas Chromatography (IGC), J. Adhes. 46 (1994) 165–180. [11] T. Hamieh, J. Schultz, New approach to characterise physicochemical properties of solid substrates by inverse gas chromatography at infinite dilution: I. some new methods to determine the surface areas of some molecules adsorbed on solid surfaces, J. Chromatogr. A 969 (2002) 17–25.
271
[12] A. Voelkel, Inverse gas chromatography in characterization of surface, Chemometr. Intell. Lab. 72 (2004) 205–207. [13] J.R. Conder, C.L. Young, Physicochemical Measurement by Gas Chromatography, Wiley-Interscience Publication, Chichester, 1979. [14] 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. [15] 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. 57 (Suppl.) (2005) S90–S91. [16] A. Voelkel, B. Strzemiecka, K. Adamska, K. Milczewska, Inverse gas chromatography as a source of physiochemical data, J. Chromatogr. A 1216 (2009) 1551–1566. [17] L. Segeren, M. Wouters, M. Bos, J. van den Berg, G. Vancso, Surface energy characteristics of toner particles by automated inverse gas chromatography, J. Chromatogr. A 969 (2002) 215–227. [18] S.K. Papadopoulou, G. Dritsas, I. Karapanagiotis, I. Zuburtikudis, C. Panayiotou, Surface characterization of poly(2,2,3,3,3-pentafluoropropyl methacrylate) by inverse gas chromatography and contact angle measurements, Eur. Polym. J. 46 (2010) 202–208. [19] N. Ahfat, G. Buckton, R. Burrows, M. Ticehurst, An exploration of interrelationships between contact angle, inverse phase gas chromatography and triboelectric charging data, Eur. J. Pharm. Sci. 9 (2000) 271–276. [20] H.E. Newell, G. Buckton, Inverse gas chromatography: investigating whether the technique preferentially probes high energy sites for mixtures of crystalline and amorphous lactose, Pharm. Res. 21 (2004) 1440–1444. [21] F. Thielmann, D.J. Burnett, J.Y. Heng, Determination of the surface energy distributions of different processed lactose, Drug Dev. Ind. Pharm. 33 (2007) 1240–1253. [22] R. Ho, S.J. Hinder, J.F. Watts, S.E. Dilworth, D.R. Williams, J.Y. Heng, Determination of surface heterogeneity of D-mannitol by sessile drop contact angle and finite concentration inverse gas chromatography, Int. J. Pharm. 387 (2010) 79–86.
