Chemical Physics 369 (2010) 126–137

8/6/2019 Chemical Physics 369 (2010) 126137

1/12

Influence of the chain length on the dynamic viscosity at high pressure of some2-alkylamines: Measurements and comparative study of some models

Masatoshi Yoshimura a,b, Christian Boned a,*, Guillaume Galliro a, Jean-Patrick Bazile a,Antoine Baylaucq a, Hideharu Ushiki b

a Laboratoire des Fluides Complexes, Facult des Sciences et Techniques, UMR CNRS 5150, Universit de Pau et des Pays de lAdour, BP 1155, 64013 Pau Cedex, Franceb Laboratory of Molecular Dynamics and Complex Chemical Physics, Department of Environmental and Natural Resource Science, Faculty of Agriculture,

Tokyo University of Agriculture and Technology, 3-5-8 Saiwai-cho, Fuchu-shi, Tokyo 183-8509, Japan

a r t i c l e i n f o

Article history:

Received 24 December 2009

In final form 12 March 2010

Available online 18 March 2010

Keywords:

2-Alkylamine

Chain length

Modeling

Pressure

Viscosity

a b s t r a c t

This work reports the dynamic viscosity data (a total of 93 points) of 2-alkylamines, which exhibit small

association, consisting of 2-aminobutane, 2-aminopentane, 2-aminoheptane and 2-aminooctane at four

temperatures between 293.15 K and 353.15 K (every 20 K), and pressures up to 100 MPa (every

20 MPa) whichallows to study the influence of thechain length. A falling-body viscometer with an uncer-

tainty of 2% was used to perform these measurements.

The variations of dynamic viscosity are discussed with respect to their behaviour due to chain length.

Seven different models, most of them with a physical and theoretical background, are studied in order to

investigate how they take the chain length effect into account through their required model parameters.

The evaluated models are based on the empirical VogelFulcherTamman (VFT) representation (com-

bined with a Tait-like equation), the rough hard-sphere scheme, the concept of the free-volume, the fric-

tion theory, a correlation derived from molecular dynamics, a model based on Eyrings absolute rate

theory combined with cubic EoS. A scaling viscosity representation has also been considered. These mod-

els need some adjustable parameters except the molecular dynamics correlation that is entirely predic-

tive. Overall a satisfactory representation of the viscosity of these 2-alkylamines is found for the differentmodels within the considered T, p range taking into account their simplicity. Moreover, it has been ver-

ified that the viscosity is a unique function of TVc where the exponent c is generally related to the steep-

ness of the intermolecular repulsive potential (T: temperature and V: specific volume).

2010 Elsevier B.V. All rights reserved.

1. Introduction

In a previous work by our group on some linear primary amines

[1], it has been underlined that there is a lack of information on the

thermophysical properties of aliphatic amines. Aliphatic amines

are used as a solvent, as a raw material in the manufacture of a

variety of other compounds, including emulsifiers and pharmaceu-

tical products. An accurate database of thermophysical properties

of these fluids is of practical and fundamental value for the chem-

ical engineering application.

With the aim of completing a database relative to this chemical

family, we have published some experimental viscosity data con-

cerning linear amines (pentyl-, hexyl- and heptylamine) versus

pressure up to 100 MPa in the temperature interval (293.15 K

and 353.15 K) [1].

In this work, the dynamic viscosity g (93 experimental data intotal) for four 2-alkylamines: 2-aminobutane (sec-butylamine,

CH3CH2CH(NH2)CH3), 2-aminopentane (CH3(CH2)2CH(NH2)CH3),

2-aminoheptane (CH3(CH2)4CH(NH2)CH3), and 2-aminooctane

(CH3(CH2)5CH(NH2)CH3), has been measured up to 100 MPa be-

tween 293.15 K and 353.15 K. The structure of these molecules is

shown in Fig. 1. To the best of our knowledge, concerning these

2-alkylamines, which have branching structure (i.e. amino group

is located at the second carbon of alkyl chain), there are no exper-

imental data of viscosity versus pressure. Mention here that the

density measurements for two 2-alkylamines (2-aminobutane

and 2-aminooctane) have previously been performed with a

vibrating-tube densimeter as a function of temperature and pres-

sure [2]. In addition, the density of 2-aminopentane and 2-amino-

heptane measured as in Ref. [2] up to 100 MPa between 293.15 K

and 353.15 K are reported in this work.

From a theoretical point of view, an accurate experimental

database of dynamic viscosity is usable to investigate the appropri-

ateness of some viscosity models taking the chain length and

branching effect into account. In particular from the structural dif-

ference between two kinds of amines, linear 1-alkylamines and

branched 2-alkylamines, it is possible to study how the difference

0301-0104/$ - see front matter 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.chemphys.2010.03.016

* Corresponding author. Tel.: +33 559 407 688; fax: +33 559 407 695.

E-mail address: [email protected](C. Boned).

Chemical Physics 369 (2010) 126137

Contents lists available at ScienceDirect

Chemical Physics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c h e m p h y s
http://dx.doi.org/10.1016/j.chemphys.2010.03.016mailto:[email protected]://www.sciencedirect.com/science/journal/03010104http://www.elsevier.com/locate/chemphyshttp://www.elsevier.com/locate/chemphyshttp://www.sciencedirect.com/science/journal/03010104mailto:[email protected]://dx.doi.org/10.1016/j.chemphys.2010.03.016


2/12

of the position of amino-group, which affects intermolecular inter-

actions, influences the macroscopic transport property i.e. viscos-

ity. In this work, seven different viscosity models are studied.

The considered models are the empirical VogelFulcherTamman

(VFT) [35] representation combined with a Tait-like equation,

the rough hard-sphere scheme [6,7], the model based on free-vol-

ume concept [810], the model based on the friction theory

[11,12], a model derived from molecular dynamics simulation

[13], the Eyring theory and cubic EoS model [14,15] and a scaling

viscosity representation [16,17].

2. Experimental procedure

The dynamic viscosity g under pressure was measured using a

falling-body viscometer. This viscometer is of the type designed

by Daug et al. [18]. The viscometer consists of two high-pressure

cells, a measuring cell and a piston cell, which are connected by a

capillary tube and a valve, see Fig. 1 in Ref. [18]. The piston cell is

connected to a pneumatic oil pump, which is used to pressurize the

viscometer. The pressure of the sample within the viscometer is

measured by an HBM-P3M manometer connected directly to the

tube betweenthe two cells, ensuring a measure of the real pressureof the sample. The pressure is measured with an uncertainty of

0.2 MPa. The temperature is measured inside the measuring cell

by a Pt100 probe connected to a classical AOIP thermometer with

an uncertainty of 0.5 K. A circulating fluid supplied by an external

thermostatic bath controls the temperature of the sample in the

measuring cell and the piston cell. The viscometer is placed in an

automated air-pulsed thermal regulator oven in order to ensure a

homogeneous temperature surrounding the system. Since the flu-

ids considered in this work are liquids at atmospheric pressure, the

filling procedure of the viscometer was done as described in Ref.

[19].

The basic principle of the falling-body viscometer is that a sin-

ker falls through a fluid of unknown viscosity under a given tem-

perature and pressure (T, p) condition. It has been emphasized by

Daug et al. [18] that, for this type of viscometer and for fluids with

a low viscosity, a working equation of the functional form

g(T, p) =f[(qS qL)Ds] should be used. This working equation re-lates the dynamic viscosity to the difference between the density

of the sinker qS and of the fluid qL, and the falling time, Ds, be-

tween two detection sensors when the velocity of the sinker is con-

stant. For fluids with a very low dynamic viscosity, such as

methane, Daug et al. [18] used a second-order polynomial in

(qS qL)Ds which implies the requirement of three reference flu-

ids in order to perform the calibration of the viscometer. In this

work the lowest viscosity is 0.285 mPa s for 2-aminobutane at

T= 353.15 K and p = 20 MPa, which is not too low, and conse-

quently it was found appropriate to use a linear relation for the

working equation as follows:

gT;p KaT;p KbT;pqS qLDs; 1which relates the dynamic viscosity to two parameters Ka and Kb. A

similar working equation has recently been used by Pensado et al.

[20].

The sinker used in this work is a solid stainless steel cylinder

with hemispherical ends and a density of 7.72 g cm3. The sinker

is designed with a ratio between its diameter and the tube diame-

ter greater than 0.98, which is substantially above the recom-

mended value of 0.93 in order to ensure a concentric fall and to

minimize eccentricity effects [2123]. Since the density of the sin-

ker is about 9 times higher than the density of the fluids considered

in this work an error in the fluid density of 0.1% results in an error

of about 1/7000 for qS qL (see relation (1)) and consequently a

very small error on viscosity. In this work, Ds

corresponds to the

average value of six measurements of the falling time at thermal

and mechanical equilibrium with a reproducibility of 0.5%.

The calibration of the viscometer has been performed using

accurate viscosity and density data for toluene and decane. The re-

quired reference viscosity and density data for toluene have been

estimated by the correlation given for the viscosity and density

by Assael et al. [24]. The reported uncertainties for the calculated

density and viscosity values are 0.03% and 2%, respectively. For

n-decane, the viscosity data has been obtained by the correlation

given by Huber et al. [25] using density values calculated by the

expression given by Cibulka and Hnedkovsky [26]. The reported

uncertainty for the calculated n-decane density and viscosity val-

ues are 0.1% and 2%, respectively. The apparatus parameters Kaand Kb are determined at each considered (T, p) condition by plot-

ting the reference viscosities of the two calibrating fluids as a func-

tion of (qS qL)Ds.In this work, the densities of two 2-alkylamines (2-aminobu-

tane and 2-aminooctane) were taken from Ref. [2], where they

have been measured up to 140 MPa between 293.15 K and

353.15 K with a vibrating-tube densimeter. The uncertainty re-

ported for these density measurements is 5 104 g cm3

(around 0.05%). Taking into account the uncertainty due to the cal-

ibration, the temperature, the pressure and the density, the overall

uncertainty for the reported dynamic viscosities is of the order of

2%. The accuracy of the device has been checked several times.

For instance, recently in our group [19], the same instrument has

been used to measure the dynamic viscosity of methanol between

293.15 K and 353.15 K up to 100 MPa. These data have been fur-

ther used in a study about a new reference correlation [27] forthe viscosity of methanol, taking into account of many methanol

literature data. Our data agree with the reference correlation with

an average absolute deviation of 1.0%, which is very satisfactory.

At atmospheric pressure (0.1 MPa) the dynamic viscosity was

obtained by measuring the kinematic viscosity, m = g/q, with a clas-sical capillary viscometer (Ubbelohde). For this purpose several

tubes connected to an automatic AVS350 Schott Gerte Analyzer

were used. The temperature of the fluid is controlled within 0.1 K

using a thermostatic bath. When multiplying the kinematic viscos-

ity with the density, the dynamic viscosity is obtained with an

uncertainty less than 1%. Each capillary tube is provided with a cal-

ibration certificate but we checked the calibration of the capillary

viscometer at several temperature using Cannon Certified Viscos-

ity Standard reference fluids.2-aminobutane (sec-butylamine, C4H11N, molar mass M= 73.14

g mol1, boiling point at atmospheric pressure Tb = 336.15 K, CAS

number 13952-84-6), 2-aminopentane (C5H13N, M= 87.16

g mol1, Tb = 363.65$ 364.65 K, CAS number 63493-28-7), 2-ami-

noheptane (C7H17N, M= mass 115.22g mol1, Tb = 415.15$

417.15 K, CAS number 123-82-0) and 2-aminooctane (C8H19N,

Fig. 1. Molecular structure of (a) 2-aminobutane, (b) 2-aminopentane, (c) 2-aminoheptane, and (d) 2-aminooctane.

M. Yoshimura et al. / Chemical Physics 369 (2010) 126137 127


3/12

M= 129.24g mol1, Tb = 438.15 K, CAS number 693-16-3) were ob-

tained from SigmaAldrich with purity of, respectively, 99%, 97%,

99% and 99% (with certificate of analysis by gas chromatography

of, respectively, 99.9%, 98.7%, 99.0% and 99.41%). These chemicals

were not subject to further purification and were directly injected

into the high-pressure cell as soon as the bottle was opened.

3. Experimental results

The values of density for 2-aminobutane and 2-aminootane ta-

ken from Ref. [2] and the values of viscosity and density measured

in this work for four 2-alkylamines are reported in Table 1. The

measurements have been performed at various temperatures

(293.15, 313.15, 333.15 and 353.15) K for pressures at (0.1, 20,

40, 60, 80 and 100) MPa. Note, however, that it was not possible

to measure viscosity or density at some conditions for 2-aminobu-

tane (p = 0.1 MPa, T= 333.15 and 353.15 K) and 2-aminopentane

(p = 0.1 MPa, T= 353.15 K) as temperature is near or over their Tb.

As we did not find in the literature viscosity values at high pres-

sure, the comparison was possible only at atmospheric pressure.

We found data only for 2-aminobutane (sec-butylamine) at

T= 303.15 K by Rao et al. [28], at T= 308.15 K by Bai et al. [29]and Subha et al. [30], and at T= 298.15 up to 323.15 K (DT= 5 K )

by Saleh et al. [31]. Among these data, it is possible to do the quan-

titative comparison only at T= 313.15 K with [31], and we obtained

a deviation of 3.9%. Fig. 2 shows for 2-aminobutane our data and

the various literature viscosity data versus temperature at atmo-

spheric pressure. The solid line corresponds to a polynomial qua-

dratic fitting obtained using all the data. For the other three 2-

alkylamines, viscosity data was not found even at atmospheric

pressure.

The variation of the viscosity versus temperature at p = 60MPa

for four 2-alkylamines of this work and two 1-alkylamines (pentyl-

amine and heptylamine) from Ref. [2] is shown in Fig. 3. Fig. 4

shows the variation of the viscosity versus pressure at

T= 313.15 K for four 2-alkylamines of this work and two 1-alkyl-

amines from Ref. [2]. From Fig. 3, it can be seen that the viscosity

decreases monotonically with increasing temperature. Fig. 4 showsthat the viscosity increases with increasing pressure. This behavior

is the one generally observed in liquids. In fact, this is not surpris-

ing as primary amines exhibit a weak hydrogen bonding. Fig. 5

shows the variation of the viscosity versus density at p = 0.1 and

100 MPa. Fig. 6 shows the variation of the viscosity versus density

at T= (293.15 and 353.15) K. It is interesting to notice that, in the

case of constant pressure (Fig. 5), the viscosity data points versus

density are roughly on the same curve independently of the alkyl

chain length of amines, however, it is not the case at constant tem-

perature (Fig. 6). This indicates that it is the density which mainly

determines the viscosity amplitude, more than the temperature or

the chain length.

The difference between four 2-alkylamines (2-aminobutane, 2-

aminopentane, 2-aminoheptane and 2-aminooctane) is the length

Table 1

Dynamic viscosity g (mPa s) and density q (g cm3) versus temperature and pressure for four 2-alkylamines (2-aminobutane, 2-aminopentane, 2-aminoheptane and 2-

aminooctane).

T (K) p (MPa) 2-Aminobutane 2-Aminopentane 2-Aminoheptane 2-Aminooctane

g q g q g q g q

293.15 0.1 0.460 0.7253a 0.622 0.7478 0.903 0.7642 1.159 0.7720a

293.15 20 0.557 0.7412a 0.731 0.7625 1.083 0.7777 1.389 0.7848a

293.15 40 0.648 0.7545a 0.864 0.7748 1.294 0.7893 1.699 0.7961a

293.15 60 0.738 0.7661a 1.001 0.7855 1.540 0.7994 2.068 0.8060a

293.15 80 0.830 0.7763a 1.141 0.7951 1.819 0.8085 2.493 0.8150a

293.15 100 0.921 0.7855a 1.285 0.8038 2.148 0.8167 3.005 0.8232a

313.15 0.1 0.360 0.7050a 0.469 0.7293 0.661 0.7477 0.829 0.7558a

313.15 20 0.440 0.7234a 0.551 0.7457 0.793 0.7627 0.981 0.7704a

313.15 40 0.512 0.7382a 0.644 0.7594 0.950 0.7753 1.187 0.7825a

313.15 60 0.584 0.7508a 0.742 0.7712 1.118 0.7863 1.417 0.7932a

313.15 80 0.657 0.7619a 0.845 0.7816 1.300 0.7961 1.673 0.8027a

313.15 100 0.730 0.7718a 0.953 0.7909 1.494 0.8049 1.955 0.8114a

333.15 0.1 b b 0.364 0.7099 0.511 0.7302 0.630 0.7396a

333.15 20 0.350 0.7052a 0.432 0.7290 0.610 0.7473 0.740 0.7555a

333.15 40 0.409 0.7220a 0.504 0.7443 0.727 0.7612 0.887 0.7690a

333.15 60 0.469 0.7358a 0.578 0.7571 0.852 0.7732 1.048 0.7806a

333.15 80 0.529 0.7480a 0.653 0.7685 0.986 0.7839 1.228 0.7909a

333.15 100 0.593 0.7586a 0.730 0.7784 1.129 0.7934 1.426 0.8000a

353.15 0.1 b b b b 0.411 0.7132 0.496 0.7232a

353.15 20 0.285 0.6871a 0.348 0.7120 0.485 0.7322 0.596 0.7416a

353.15 40 0.336 0.7058a 0.410 0.7290 0.576 0.7477 0.708 0.7562a

353.15 60 0.386 0.7212a 0.471 0.7430 0.673 0.7607 0.825 0.7687a

353.15 80 0.436 0.7342a 0.531 0.7552 0.778 0.7721 0.949 0.7797a

353.15 100 0.488 0.7457a 0.590 0.7659 0.891 0.7822 1.080 0.7895a

a

The values for 2-aminopentane and 2-aminoheptane are taken from Ref. [2].b Viscosity and density cannot be measured on the condition in empty columns as temperature is near or over Tb (p = 0.1 MPa).

Fig. 2. Comparison with literature data of viscosity at atmospheric pressure for 2-

aminobutane. }: Bai 2005 [29], s: Rao 2006 [28], h: Saleh 2001 [31], 4: Subha

2004 [30], d: this work. Solid line is fitting curve for all the values from literature

and this work with quadratic polynomial function.

128 M. Yoshimura et al. / Chemical Physics 369 (2010) 126137


4/12

of alkyl chain in their molecular structure (Fig. 1) which affects the

dynamic viscosity for a given temperature and pressure. As ex-

pected the viscosity increases with the chain length for a given

set of T and p (Figs. 3 and 4), but notice that it is less sensitive to

the chain length for a given set of p and q or T and q (Figs. 5 and6). Further, as observed in the density measurement [2], 2-alkyla-

mine which has longer alkyl chain tends to form denser, less com-

pressible liquid than other 2-alkylamines which have shorter alkyl

chain at a given pressure.

From the viewpoint of the difference in the position of amino

group, Figs. 3 and 4 indicate that 1-alkylamines, which have linear

structure, show higher viscosity than 2-alkylamines, which have

branching structure, in the temperature and pressure range con-

sidered here. The same tendency was observed in the measure-

ment of density [2].

Generally, when a fluid is brought under pressure (compressed),

the flexibility and mobility of the molecules are reduced, since the

distance and space between the molecules become shorter, result-

ing in a reduction of the fluid mobility and an increase in the vis-

cosity. Kioupis and Maginn [32] explained the fact that the

viscosity increases with pressure as a result of a reduction in the

liquid void volume coupled with the molecular structure, resulting

in a lower motion of the molecules, because the motion is related

either to molecules jumping or forcing adjacent molecules into

these voids. Therefore, when the number of voids decreases with

increasing pressure, complex molecules with a low flexibility will

have difficulties of making these jumps or forcing other molecules

into these voids, resulting in the trapping of the molecules and a

higher viscosity.

For the four amines considered in this work, a slightly sharper

increase of dynamic viscosity against pressure has been found for

2-aminooctane than for 2-aminoheptane, -pentane and -butane

(Fig. 4) due to the increase of the alkyl chain length, which results

in an important reduction of the fluid mobility (higher viscosity)

when brought under pressure.

4. Discussion

Since these molecules have a different chain length, it is inter-

esting to investigate how the effect of the chain length is taken into

account by some viscosity models. First we consider an empirical

model based on VogelFulcherTammans (VFT) viscosity repre-

sentation [35] combined with Tait-like equation. Second, six more

or less recently derived models with a physical and theoretical

background are considered: rough hard-sphere scheme [6,7],

free-volume scheme [810], friction theory [11,12], correlation

based on molecular dynamics simulation of LennardJones fluid

[13], a scaling viscosity model [16,17], and model based onthe Eyr-

ing theory combined with a cubic EoS [14,15]. For the models using

experimental density values, they are taken from Ref. [2] and fromthis work (Table 1).

Fig. 3. Dynamic viscosity versus temperature at p = 60 MPa for four 2-alkylamines

(j: 2-aminobutane, : 2-aminopentane, N: 2-aminoheptane and d: 2-aminooc-

tane) and two 1-alkylamines (}: pentylamine and 4: heptylamine) [2].

Fig. 4. Dynamic viscosity versus pressure at T= 313.15 K for four 2-alkylamines (j:

2-aminobutane, : 2-aminopentane, N: 2-aminoheptane and d: 2-aminooctane)

and two 1-alkylamines (}: pentylamine and 4: heptylamine) [2].

Fig. 5. Dynamic viscosity versus density at p = 0.1 MPa (j: 2-aminobutane, : 2-

aminopentane, N: 2-aminoheptane and d: 2-aminooctane) and p = 100 MPa (h: 2-

aminobutane, }: 2-aminopentane, 4: 2-aminoheptane and s: 2-aminooctane).

Fig. 6. Dynamic viscosity versus density at T= 293.15K (j: 2-aminobutane, : 2-

aminopentane, N: 2-aminoheptane and d: 2-aminooctane) and T= 353.15K (h: 2-

aminobutane, }: 2-aminopentane, 4: 2-aminoheptane and s: 2-aminooctane).



5/12

In order to make a comparative study of the performance of dif-

ferent viscosity models to represent the viscosity, the following

definitions are used.

Deviationi 1 gcalc;i=gexp;i;

AAD 1NPX

NP

i1jDeviationij;

MD MaximumjDeviationij;

Bias 1NP

XNPi1

Deviationi;

2

where NP is the number of data points, gexp the experimental vis-

cosity and gcalc the calculated viscosity. The AAD (average absolute

deviation) indicates how close the calculated values are to the

experimental values, and the quantity Bias indicates how well the

experimental points are distributed around the calculated points.

If Bias is equal to AAD, then all of the calculated values are below

the experimental values. Further, the quantity MD refers to the

absolute maximum deviation.

4.1. VogelFulcherTammanTait representation

The first viscosity model considered in this work is one already

proposed and described more precisely in a previous work of our

team [33]:

gp; T A expB=T C exp D ln ET pET 0:1MPa

; 3

where D was assumed to be temperature independent and E(T) is a

second-order polynomial as E(T) = E0 + E1T+ E2T2. This equation is

derived from a Tait-like equation combined with the empirical Vo-

gelFulcherTammans model [35] which can be considered as

Andrades modified equation [34].

Experimental viscosity data have been fitted with Eq. (3) by

LevenbergMarquardt algorithm and the obtained coefficients

are shown in Table 2 for four 2-alkylamines. In order to use them,p is in MPa, Tin K, Min kg m3 and the viscosity is in mPa s. Notice

that the worse overall MD, found for 2-aminooctane at T= 353.15 K

and p = 0.1 MPa, is of the same order of magnitude as the experi-

mental error.

4.2. The rough hard-sphere scheme

The second viscosity model employed is based on the fact that a

corresponding states relationship exists [6,7] between the experi-mental transport properties of rough non-spherical molecules

and the smooth hard-sphere values (subscript shs):

gexp Rggshs; 4where the proportionality factor Rg, described as the roughness fac-

tor, accounts for the roughness and non-spherical shape of the mol-

ecule. The roughness factor is related to the non-spherical shape of

the molecule (see e.g. [35]) and Rg = 1 corresponds to a spherical

molecule. Dymond and Awan [6] derived the following expression

relating the reduced smooth hard-sphere viscositygshs to the exper-

imental value gexp:

gshs gexpRg

6:035 108 ffiffiffiffiffiffiffiffiffiffi1

MRTrgexpv

2=3

Rg5

with v the molar volume, Mthe molar mass and R the gas constant.

In order to determine Rg and v0, the close packing molar volume, for

a given temperature a plot of log10gexp versus log10(m) from the

experiment is superimposed on a universal plot of log10gshs versus

log10(m/m0) from the hard-sphere theory by vertical and horizontal

adjustments. The empirical expression for this curve is:

log10gexpRg

X7i0

ag;i1

vr

i: 6

The ag,i coefficients [7] are universal, independent of the chemical

nature of the compound. This has been verified by Baylaucq et al.

[36,37]. Further, it is generally assumed that v0 is temperature

dependent, whereas Rg is temperature independent [35].

The estimated hard-core volume v0 for each temperatureand the roughness factor Rg are given in Table 3 for the four

Table 2

Values of coefficients and results obtained with the VogelFulcherTammans representation (Eq. (3)).

2-Aminobutane 2-Aminopentane 2-Aminoheptane 2-Aminooctane Overall

A 0.00017525 0.017364 0.011448 0.047302

B 4941.2 835.53 1213.6 561.81

C 334.21 59.182 14.627 116.71

D 0.93693 1.5163 3.5456 7.1394

E0 17.154 165.64 1491 2517

E1 0.58743 1.8448 10.84 17.388

E2 0.0011376 0.002501 0.015464 0.021993

AAD (%) 0.250 0.410 0.541 0.736 0.484

Bias (%) 0.000514 0.00411 0.0338 0.0539 0.0228MD (%) 0.76 0.97 1.16 2.64 2.64

Table 3

Adjusted values of the roughness factor Rg and the hard-core volume v0 (m3 mol1) in the rough hard-sphere viscosity scheme (Eqs. (5) and (6)).


Rg 1.261 1.287 1.241 1.290

v0 (293.15 K) 6.445E05 7.730E05 1.049E04 1.190E04

v0 (313.15 K) 6.378E05 7.621E05 1.036E04 1.174E04

v0 (333.15 K) 6.301E05 7.506E05 1.025E04 1.160E04

v0 (353.15 K) 6.198E05 7.394E05 1.014E04 1.147E04

AAD (%) 1.033 0.631 0.628 1.092 0.846

Bias (%) 0.086 0.036 0.053 0.050 0.005

MD (%) 2.07 1.50 1.34 2.77 2.77

130 M. Yoshimura et al./ Chemical Physics 369 (2010) 126137


6/12

2-alkylamines. The calculation has been done in order to minimize

MD for all the data. The modeling for the four compounds resulted

in an AAD of 0.85% with a MD of 2.77% close to the magnitude of

the experimental error. The obtained v0 is increasing with molar

mass and is decreasing with temperature. The obtained Rg is not

regular against molar mass but as the values are comprised be-

tween 1.24 and 1.29 this fact is perhaps not significant. Compared

with the case of 1-alkylamines [1], it seems that Rg

is less impor-

tant for 2-amines than for 1-amines (1.394 for 1-heptylamine

and 1.241 for 2-aminoheptane). This result is probably due to the

branching effect.

In order to make an analysis of the influence of the molecular

structure, the next modeling step is to represent the variation of

the dynamic viscosity as a function of molecular mass. As the var-

iation of Rg is not simple, the parameter v0 has only been corre-

lated, by expressing it with a simple linear function, on the

overall data set (93 points). The correlation used for (m0(T, M) isthe following:

m0T;M a1M a0T b1M b0: 7This expression allows a reduction of the total number of parame-

ters from20 to 8. Table 4 shows parameters and result of re-estima-

tion using the value of Rg obtained in Table 3. This parameter

reduction resulted in an AAD of 4.37% with a MD of 8.80%. Both

AAD and MD have increased and are higher than the magnitude

of experimental uncertainty. There seems to be branching effect,

which cannot be described very well with the simple linearization

of parameters against temperature and molar mass. This model,

with the development of parameters Rg and v0, has already been

successfully applied in the case of 1-alkylamines [1] and in some

other previous works [38,39].

4.3. Free-volume viscosity model

Based on the free-volume concept, an approach has recently

been proposed in order to model the viscosity of Newtonian fluids

in the gaseous and dense states [8,9]. In this approach, the total vis-cosity g can be separated into a dilute gas viscosity term g0 and anadditional termDg, in the following way:

g g0 Dg: 8The termDg characterizes the passage in the densestate and is con-

nected to the molecular structure via a representation of the free

volume fraction. The general expression of the free-volume viscos-

ity model is [8]:

g g0 qaq pM=qffiffiffiffiffiffiffiffiffiffiffiffiffi

3RTMp

exp B

aq pM=qRT

3=2" #; 9

where corresponds to a characteristic molecular length.a is linked

to the barrier energy E0 = aq, which the molecules have to exceed inorder to diffuse. B is characteristic of the free-volume overlap. This

equation involves three physical parameters , a and B, which are

characteristic of the molecule. This model has been shown to accu-

rately represent the viscosity behavior of various hydrocarbons over

wide ranges of temperature and pressure in the gaseous, liquid and

dense supercritical states. Recently [10], the model has been gener-

alized to the simultaneous modeling of the self-diffusion coefficient

and dynamic viscosity at high pressure (up to 500 MPa).

The dilute gas viscosity term g0 can be obtained by any appro-

priate model, for instance the model by Chung et al. [40] which is

able to predict the dilute gas viscosity of several polar and non-po-

lar fluids within an uncertainty of 1.5%. The model is an empirical

correlation derived from the ChapmanEnskog theory [41] and the

reduced collision integral expression for the LennardJones 12-6

potential of Neufeld et al. [42]. This model is related to the critical

temperature Tc, the critical molar volume vc and the acentric factor

x.

Unfortunately, experimental critical property data of organic

compounds are limited due to the fact that many compounds be-

come unstable during measurements near or even far from the

vicinity of the critical point. We found some experimental and rec-

ommended values in Marsh et al. [43], which is a part nine of re-

view series for vaporliquid critical properties of elements and

compounds. For 2-aminobutane (i.e. sec-butylamine)

Tc = 514.3 0.3 K (recommended value is 514 1 K) andpc = 5.0 MPa (recommended value is 5.0 0.5 MPa), and for 2-ami-

noheptane Tc = 598.0 0.3 K (recommended value is 598.0 0.6).

In such conditions, mathematical models can be used to provide

a reasonable estimate of these properties. A variety of estimation

methods for critical property data are available in the open litera-

ture. A broad overview of these methods together with a detailed

discussion of their reliability was given by Poling et al. [44]. Re-

cently, Nannoolal et al. [45] developed a new group contribution

method for the prediction of critical properties and their method

has been compared with 10 well-known estimation methods from

literature. More recently, a position group contribution method,

which requires only the knowledge of their chemical structure,

for the prediction of critical temperature Tc [46], critical pressure

pc [47] and critical volume vc [48] have been proposed. Their meth-

od, owing to the utilization of the position compensation factor,

demonstrates significant improvements compared to the previ-

ously used first- or second- order method, especially in the capabil-ity of distinguishing between isomers. In order to evaluate the

dilute gas viscosity term g0, in this work, the critical propertieswere estimated by the position group contribution method [46

48]. In addition, the acentric factor was estimated by the method

recommended in Ref. [44], in which the estimated critical proper-

ties Tc and pc and the experimental Tb were used with the recom-

mended equation:

x lnPc=1:01325 f0Tbr

f1Tbr ; 10

where Tbr Tb=Tc;f0 and f1: analytical expressions developed by

Ambrose and Walton [49]. Table 5 shows the values ofTb described

as material specification by SigmaAldrich and estimated values ofcritical properties and acentric factor. As Tb of 2-aminopentane and

2-aminoheptane are described as ranged value, we used the middle

value of the range. By the comparison with experimental value, we

found deviations for 2-aminobutane of Tc and pc are 1.4 and 9.1%,

and a deviation for 2-aminoheptane of Tc is 0.70%.

Table 4

Parameters and results of the rough hard-sphere viscosity scheme with the linearization of parameter v0 by Eq. (7).

v0 (m3 mol1) a0 a1 b0 b1

6.463E09 4.970E07 4.225E06 1.108E03


AAD (%) 7.01 6.53 1.72 2.22 4.37

Bias (%) 7.01 6.53 1.72 2.09 1.07

MD (%) 8.70 8.80 3.88 8.80 8.80



7/12

It is important to notice that, as the compounds studied here are

in the dense state, the value ofg0 is low compared to the total vis-cosity. For the four compounds the maximum value for dilute gas

viscosity is g0 = 0.008 mPa s for 2-aminobutane at T= 353.15 K (to-

tal viscosity of 2-aminobutane at p = 20 MPa and T= 353.15K is

g = 0.285 mPa s). We noticed also that g0 is not very sensitive tosmall variations of the critical parameters and acentric factor.

The three characteristic parameters in Eq. (9) have been esti-

mated by minimizing MD for each of the four amines. The esti-

mated parameters are presented in Table 6 with the deviation

results. The dynamic viscosity is obtained in (Pa s), when all vari-

ables and properties are inserted in SI units. It should be noticed

that the AAD for each of the compounds are lower than the exper-

imental uncertainty with MD lower than 3.30% (obtained for 2-

aminooctane at p = 20 MPa and T= 313.15 K). The overall results,

with an AAD of 1.01%, can be considered very satisfying because

the calculations involve only three parameters for each one of

the four amines. A comparison of the specific energy parameter ashows that it increases with the number of CH2 groups. By multi-

plying the specific energy parameter with the density, an influence

on the energy barrier, E0 = aq, due to the chain length effects canbe seen, since the energy barrier E0, which the molecules has to ex-

ceed in order to diffuse, is approximately 7378% higher for 2-ami-

nooctane than for 2-aminobutane. The larger energy will result in a

lower mobility of the molecules in 2-aminooctane and conse-

quently in a higher viscosity. In case of the B parameter, which isrelated to the free-volume overlap of molecules, a pronounced ef-

fect due to the chain length effect is also found: it decreases with

the chain length. The obtained B value of 2-aminooctane is signif-

icantly lower than for 2-aminobutane. The effect of chain length

can also be seen on the characteristic molecular length , which

is smaller for 2-aminooctane (0.04085 nm) than for 2-aminobu-

tane (0.06391 nm). This is related to the molecular structure that

is longer for 2-aminobutane molecule (Fig. 1). According to [8],

in Eq. (9), L2=bf, where L2 is an average characteristic molecular

quadratic length, and bf is the dissipation length of the energy

E= E0 +pM/q where the term pM/q =pv is related to the energy

necessary to form the vacant vacuums available for the diffusion

of the molecules. Certainly as L2 seems to increase with the length

of alkyl chain, the decrease of means that bf increases with alkyl

chain length more than L2. Finally, it has to be mentioned that it is

the value for the molecular energy which has a more important

contribution to the total viscosity in excess of the dilute gas viscos-

ity. This is confirmed by the fact that 2-aminobutane, which has

lowest experimental viscosity in the four compounds, shows lower

value for the energy parameter a than the other compounds,though it shows higher value of and B than the other compounds.

Compared with the case of 1-alkylamines [1], no particular effect

appears that could clearly be related to the branching effect.

The procedure presented for the rough hard-sphere scheme can

also be used to reduce the number of the parameters. Each main

parameter ;a and B has then been correlated against the numberof carbon atoms n of the alkyl chain with linear equations as

A =A1M+A0 (with A ;a or B) so that only six parameters areused, for the four amines. The results and parameters obtained

by this correlation are shown in Table 7. Compared with the previ-

ous result (Table 6), AAD shows a little bit worse result (1.62% for

all the data instead of 1.01%) but it is still lower than the experi-

mental uncertainty. For the maximum deviation the result is

3.68% for the 2-aminobutane (T= 293.15 K, p = 0.1 MPa) and 2-

aminooctane (T= 293.15, p = 100 MPa). It should be noticed that

a increases with M, and that and B decrease when M increases,as already observed in Table 6.

In Eq. (9) the density appears explicitly and for the calculation

we used the experimental density data. It is important to under-

line, however, that the model could be applied even if the density

is not known by using an efficient equation of state along with Eq.(9). Of course the parameters obtained in such way are partially

linked to the chosen equation of state. Notice that this remark is

valid for all the models where density appears explicitly. In this

aim, recently the free-volume model has been successfully coupled

Table 5

Experimental boiling point (at atmospheric pressure) and evaluated values of critical

properties by the position group contribution method [4648] and acentric factor by

the method in Ref. [44].

Tb (K) Tc (K) pc (MPa) vc (cm3 mol1) x

2-Aminobutane 336.15 521.4 4.547 274.0 0.2757

2-Aminopentane 364.15 547.5 4.190 333.1 0.3683

2-Aminoheptane 416.15 593.8 3.671 455.6 0.5654

2-Aminooctane 438.15 615.3 3.466 518.0 0.6307

Table 6

Results obtained on the four compounds with the free-volume viscosity model.

Parameters 2-Aminobutane 2-Aminopentane 2-Aminoheptane 2-Aminooctane Overall

(nm) 0.0639102 0.0442546 0.0469946 0.0408502

a (J m3 kg1 mol1) 93.766 125.364 140.117 160.181

B 0.00827095 0.00675202 0.00657969 0.00613258

AAD (%) 0.90 0.56 0.77 1.82 1.01Bias (%) 0.051 0.217 0.071 0.555 0.090

MD (%) 1.81 1.46 1.53 3.30 3.30

Table 7

Results obtained on the four compounds with the free-volume viscosity model and the correlated parameters (A = A1 M + A0 with M in kg mol1).

A1 A0

(nm) 0.0428623 0.0526912

a (J m3 kg1 mol1) 613.256 67.3793

B 0.00112600 0.00688086


AAD (%) 2.08 1.13 1.32 1.94 1.62

Bias (%) 1.506 0.913 1.191 0.177 0.263

MD (%) 3.68 3.55 3.15 3.68 3.68



8/12

with the statistical associating fluid theory (SAFT model) [50].

Moreover in this latter work the authors successfully demonstrate

that it is possible to correlate and predict simultaneously the vis-

cosity ofn-alkanes using the free-volume model with SAFT theory,

with parameters that are universal for the whole series of

n-alkanes with only six parameters. In the data-pool fitting, they

assumed, like in the present work, that the parameters of the

free-volume model scale linearly with molecular weight, and this

simple assumption has been effective.

4.4. f-Theory model

Starting from basic principles of mechanics and thermodynam-

ics, the friction theory (f-theory) for viscosity modeling has been

introduced [11]. In the f-theory the total viscosity can be written

as:

g g0 gf 11

where g0 is the dilute gas viscosity and gf the residual friction con-

tribution. The friction contribution is related to the van der Waals

attractive and repulsive pressure terms, pa and pr, of an equationof states (EoS), such as the Peng and Robinson (PR) [51] or the

SoaveRedlichKwong (SRK) [52] ones. Based on this concept, a

general f-theory model [12] valid for hydrocarbons has been intro-

duced with 16 constants identical for all hydrocarbons. In this work,

the f-theory approach in conjunction with the PR EoS, as described

in our previous work [53] has been used. In this approach there are

two adjustable parameters, a characteristic critical viscosity gcand a third order constant d2. The required dilute gas viscosity

of the pure compounds has been obtained by the Chung et al. model

[40], as for free-volume theory.

The two adjustable parameters, the estimated critical viscosity

gc and the third order friction constant as well as the deviation re-sults are shown in Table 8. The obtained results are satisfactory

taking into account that they are obtained in conjunction with a

simple cubic EoS with only two adjustable parameters. In compar-

ison with the two previous other models with a physical back-

ground (rough hard-sphere scheme and free-volume theory)

Table 8, shows that the overall values of AAD and MD found for this

model are slightly higher.

Plotting the critical viscosity reported in Table 8 as a function of

the molar mass reveals that the critical viscosity is nearly a linear

function of the molar mass (of the form gc = gc1M+ gc0) The thirdorder friction constant reported in Table 8 cannot be represented

by a linear relationship with the molar mass.

By suggesting that the critical viscosity is a linear function of

the molar mass and assuming that the third order friction constant

is a constant independent of the 2-aminohydrocarbon, the three

adjustable parameters were obtained and reported in the table

caption of Table 9. The deviation results obtained are reported in

Table 9. The overall AAD = 3.08% with Bias= 0.35% and

MD = 10.4%, which is a satisfactory result.

It is important to note that when accurate and reliable viscosity,

density and phase behaviour predictions can be achieved, then

accurate models for these properties can be linked and connected.

In this sense, as already underlined above, the free-volume model

has been successfully coupled with the statistical associating fluid

theory for pure compounds [50]. The friction theory has also been

coupled to different types of SAFT models [50,54]. In the f-theory

the density does not appear explicitly, but it is necessary to know

the attractive pressure and the repulsive pressure. In a recent pa-

per it is shown how the f-theory can be linked to practically any

type of EoS [55] ranging from highly theoretical EoS to highly accu-

rate empirical reference EoS. However, the adjustable parameters

in this model are linked to the equation of state chosen for the

calculation.

4.5. Molecular dynamics viscosity model

Recently, a predictive viscosity approach has been introduced

for simple pure fluids and mixtures over a wide range of tempera-

ture and pressure [13]. This approach is derived from molecular

dynamics simulations using a corresponding state scheme, where

the LennardJones (LJ) fluid is taken as the reference compound

and a one-fluid approximation is applied to mixtures. A simple cor-

relation has been developed in order to accurately reproduce re-

cent molecular dynamics results on the LJ fluid over a large

range of thermodynamic states [13].

In this model, the reduced viscosity g* has been expressed ver-

sus reduced temperature T* and reduced density q* (see [13] for

details) as a sum of a classical ChapmanEnskog dilute density

contribution g0 and of a residual viscosity contribution Dg [13] asin a free-volume and f-theory approaches:

gT;q g0T DgT;q: 12In this model the dilute density contribution is defined as

g0T 0:17629AcffiffiffiffiffiT

p

Xv; 13

where Ac = 0.95, andXv is the collision integral. Neufeld et al. [42]

have derived expressions for different collision integrals. In Ref.

Table 8

Results obtained with the f-theory viscosity model in conjunction with the PR EOS by adjusting the critical viscosity gc and the third order friction constant d2.


gc (lP) 346.14 331.71 301.41 297.309d2 (lP/bar

3) 4.69969 109 5.21001 109 3.58445 109 3.81666 109

AAD (%) 1.61 1.09 3.43 3.97 2.56

Bias (%) 0.04 0.02 0.15 0.21 0.11

MD (%) 4.75 3.50 7.45 8.22 8.22

Table 9

Results obtained with the f-theory viscosity model in conjunction with the PR EOS by correlating the critical viscosity gc = gc1 M + gc0 with M in kg mol1 [gc1 = 1173.36

(lP mol kg1) and gc0 = 437.693 (lP)] and the third order friction constant d2 = 4.26789 109 (lP/bar3).


AAD (%) 1.68 2.06 4.44 3.99 3.08

Bias (%) 0.70 1.27 3.00 1.07 0.35

MD (%) 6.26 4.89 10.4 8.74 10.4



9/12

[13] the expression for the 12-6 collision integral is used. The resid-

ual viscosity contribution is expressed as:

DgT;q b1expb2q 1 b3expb4q 1 b5expb6q 1T2; 14

where the bi coefficients have been regressed against molecular

dynamics simulations results on the LJ pure fluid [13].In order to apply this approach to real fluids, the two LJ molec-

ular parameters (rii: molecular length and eii: energy parameter)that are supposed to represent the real compound, are required.

These molecular parameters have been related to the critical tem-

perature Tc in K and the critical molar volume vc in m3 mol1,

through

eii kBTc1:2593

; 15

rii 0:302 vcNA

1=3; 16

Eq. (15) has been proposed by Chung et al. [40] and Eq. (16) has

been proposed by Galliro et al. [13]. The unit for eii is (J), and for

rii is (m).

It is worth to underline that this model is entirely predictive,

contrary to the models described in previous sections, as no

parameter adjustment has to be done to the experimental viscosity

database. Using the critical properties given in Table 5 we found

the results indicated in Table 10. The model underestimates the

viscosity. The results are much worse than the ones from the pre-

vious methods, but this model is the only one purely predictive asthere is no parameter to adjust. Nevertheless, it is important to no-

tice that the model predicts correctly an increase of the viscosity

when the pressure increases and a decrease of the viscosity when

the temperature increases. Moreover the viscosity increases from

2-aminobutane to 2-amoinooctane (see Fig. 7 which correspond

to p = 60 MPa). From the viewpoint of the difference of molecular

structure, the overall results for branching amines of this work

are approximately twice worse than the ones for linear amines

studied in previous work [1]. The increase of the difference from

the theory value is thought to be due to the branching effect,

namely, this suggests the insufficiency of this method in order to

predict the structural isomers.

It is possible to improve the results with adjustment of the

molecular parameter r, but in this case the model is no more pre-dictive. In fact, it should be mentioned that Eq. (16) in conjunction

with the critical molar volume vc is efficient only for simple com-

pounds [13]. For more complex molecules, vc (i.e. r, see Eq. (16))

should be adjusted on viscosity data. The values of vc adjusted by

minimizing MD and the results are shown in Table 11. By perform-

ing this adjustment in order that MD would be minimum, we

found, for 2-aminobutane, -pentane, -heptane and -octane, respec-

tively, the adjusted values vc = 295.50, 350.75, 468.47 and

529.29 cm3 mol1 instead of the estimated values vc = 274.0,

333.1, 455.6 and 518.0 cm3 mol1 (Table 5). The AAD, Bias and

MD have been significantly improved in comparison with the ones

in Table 10. This direct estimation of vc for the four compounds

using experimental viscosities clearly indicates that the model is

very sensitive to the LJ parameter rii as previously shown [13].Finally we have supposed that the molecular length r has the

relationship with molar mass as r = r1M+ r0. The correlated

parameters for r, the values ofvc calculated by Eq. (16) and results

are presented in Table 12. The AAD, Bias and MD are worse than

the ones obtained by the estimation ofvc for each compounds (Ta-

ble 11), however, they are still much better than the ones obtained

by the purely predictive method (Table 10).

As already underlined, in a predictive way this scheme provides

a reasonable estimation of the viscosity of these compounds. The

overall results can be considered satisfactory compared to the sim-

plicity of this scheme. An improvement is obtained with only 1 ad-

justed molecular parameter per compound (or 2 for the four

compounds taken simultaneously), the critical molar volume of

the pure fluids (or the molecular size).

4.6. Eyring theory and cubic EoS model

On the basis of the Eyrings Absolute Rate Theory [56], Lei et al.

[57] derived the following two-parameters model for correlating

the viscosity of pure liquids under saturated conditions:

Table 10

Results obtained with the molecular dynamics viscosity model.


AAD (%) 42.3 33.9 21.7 17.7 28.9

Bias (%) 42.3 33.9 21.7 17.7 28.9

MD (%) 46.4 39.5 29.9 30.8 46.4

Fig. 7. Molecular dynamics based correlation: comparison between the experi-

mental (closed symbols) and the estimated (open symbols) dynamic viscosity at

p = 60 MPa for 2-aminobutane (j, h), 2-aminopentane (h, }), 2-aminoheptane (N,

4) and 2-aminooctane (d, s).

Table 11

Results obtained with the molecular dynamics viscosity model with the adjusted critical molar volumes.


vc (cm3 mol1) 295.50 350.75 468.47 529.29

AAD (%) 4.24 2.94 3.90 5.05 4.03

Bias (%) 2.27 0.83 3.06 3.59 1.30

MD (%) 8.44 6.07 5.95 8.88 8.88

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


10/12

g RTVL

1

cexp a

DUvapRT

exp ZL; 17

where c (s1) represents the frequency of activated molecule flow

to vacancy site, the reciprocal of which can be considered the mean

free residence time as sR = 1/c, a is the proportionality factor be-

tween the activation energy DG and the internal energy of vapor-

ization DUvap, VL is the volume of the liquid, and ZL is the

compressibility factor of the liquid, respectively. On the other hand,

in order to extend the applicability of Eq. (17) to a wider tempera-

ture range, Macas-Salinas et al. [14,15] adopted successfully

power-law dependence betweenD

G

andD

Uvap and their three-parameters viscosity model is written as

g RTVL

1

cexp a

DUvapRT

b" #expZL: 18

In this work we use a slightly different equation than Eq. (18).

The viscosity model of the Eyring type for pure liquids is written

as:

g RTVL

1

cexp a

DUvapRT

expbZL: 19

This equation has still 3 adjustable parameters, but keep the linear

variation of ln(g) versusDUvap. It is worthwhile to say here that this

relation has been privately suggested to us by Prof. Macias-Salinas

et al. [14,15]. The equilibrium properties VL, DUvap = Uvap UL andZL in Eq. (19) at given temperature and pressure are computed from

a cubic PR EoS [51].

The three parameters of Eq. (19) were estimated for the four 2-

alkylamines considered in this work. The correlating results are

summarized in Table 13. The performance of Eq. (19) in correlating

saturated liquid viscosity of associated fluids was remarkably good

with overallAAD values of 0.94% using Eyring-PR model, whose va-

lue is within experimental uncertainty. For the maximum devia-

tion, the result is 5.45% for 2-aminooctane (T= 293.15 K,

p = 100 MPa). In addition, the influence of the molar mass on

parameters was modeled supposing that the parameters are linear

function of the molar mass as A =A1M+A0 (withA = a or b or c). Asshown in Table 14, AAD shows a little bit worse result compared

with the previous result (overall 1.29% instead of 0.94%) but it is

still within experimental uncertainty. For the maximum deviation,

the result is 6.00% for 2-aminooctane (T= 293.15 K, p = 100 MPa).

The overall results of this model shown in Tables 13 and 14 are

very well, compared to other viscosity models used in this work.From the viewpoint of the estimated values of characteristic

parameters, taking into account of the results of Lei et al. [57]

and Macas-Salinas et al. [14,15], there is no clear correlation be-

tween molecular shape and the value of the parameters. In addi-

tion, even if it is the same fluid, the values of the parameters are

different depending on the model expression and the type of EoS

used to calculate the equilibrium properties. However, as a satis-

factory result was obtained in this work by linearizing the param-

eters against molar mass, it is possible to consider that the value of

parameters reflects the difference of the molecular motion that

originates in the difference of the molecular structure (length of

the carbon chain). Focusing on the parameterc for the four 2-alkyl-

amines used here, the molecule which has the longer carbon chain

shows lower value ofc, i.e., lower frequency of the activated mol-ecule displacement to the vacancy site. Notice that when applied to

linear amines, pentylamine, hexylamine and heptylamine [1] this

model gives also a very good representation. The worst case is

for heptylamine : AAD = 1.26%, Bias = 0.64% and MD = 4.0%. No par-

ticular effect appears that could clearly be related to the branching

effect.

Table 12

Results obtained with the molecular dynamics viscosity model and the correlated parameters (r = r1M+ r0 with M in kgmol1).

r (nm) r1 r02.023440 0.381871


vc (cm3 mol1) 296.65 346.89 463.87 531.06

AAD (%) 4.08 8.60 6.79 8.02 6.87

Bias (%) 0.90 8.60 6.79 7.50 1.75

MD (%) 13.0 12.7 13.0 13.0 13.0

Table 13

Results obtained with the model based on Eyring theory and PR EoS.


a 0.26408 0.26399 0.22006 0.22040

b 0.11446 0.10304 0.10160 0.09770

c 1011 (s1) 9.1115 9.2957 6.3074 6.2322

AAD (%) 1.01 0.51 0.56 1.69 0.94

Bias (%) 0.020 0.001 0.010 0.043 0.019

MD (%) 2.69 1.56 2.40 5.45 5.45

Table 14

Results obtained with the model based on Eyring theory and cubic EoS and the correlated parameters (A = A1M+ A0 with M in kg mol1).

A1 A0

a 1.5028 0.39482

b 0.21405 0.12609

c 1011 (s1) 115.67 19.5938


AAD (%) 1.44 1.10 0.740 1.87 1.29

Bias (%) 0.162 0.278 0.324 1.46 0.393

MD (%) 5.16 3.24 1.85 6.00 6.00



11/12

4.7. Roland model for viscosity scaling

Recently [16], it has been put into evidence the phenomenon of

superposition of the relaxation times, s, for various glass-forming

liquids and polymers when expressed as a function of TVc (V: spe-

cific volume V= 1/q, and c a constant characteristic of the mate-rial). Roland et al. [17] have extended this thermodynamic

scaling to the viscosity of several real fluids. For a given compound

the viscosity g is only a functional of the quantity TVc

gT;V gT;q fTVc fTqc: 20According to Refs. [16,17,58] the parameter c reflects the mag-

nitude of the intermolecular forces. It links the thermodynamic to

the transport property (here viscosity). In Ref. [17], Eq. (20) has

been verified for several liquids. Eq. (20) has been later verified

by other authors. In particular in Refs. [59,60] the superposition

is very clearly observed for several pentaerythritol ester lubricants,

for linear and branching alkanes, polar liquids, ionic liquids, and

alcohols.

We have estimated the exponent c for the four 2-alkylaminesconsidered in this work. In order to model the influence of the mo-

lar mass in this narrow molar mass interval, we made fittingimposing c = c1M+ c0. The function f(TV

c) in Eq. (20) is not speci-

fied, however, using polynomial function of (TVc) we obtained a

very satisfactory scaling result as c =0.018852 M+ 9.411908(M in g mol1). For each 2-alkylamine the viscosity data collapse

onto a single master curve. In fact as shown in Fig. 8, there is a sin-

gle master curve identical for the four amines (c = 8.03, 7.77, 7.24and 6.98 for 2-aminobutane, 2-aminopentane, 2-aminoheptane

and 2-aminooctane, respectively). Fig. 8 shows really a very good

superposition of these four 2-alkylamines, which have branching

structure. It is then clear that Roland et al. [16,17] scaling relation

can be used to determine the volume and temperature dependence

of viscosity, over broad temperature and pressure range. Neverthe-

less, for some strongly hydrogen-bonded materials (water, low

molecular weight polypropylene glycol), the superpositioning fails[17]. Despite the small association of these branching amines, the

representation makes sense as Eq. (20) reproduces the experimen-

tal viscosity very well. According to Eq. (20), scaling factor c corre-sponds to the weight on V= 1/q, the magnitude of the c valuemeans the magnitude of a relative influence of the density on the

viscosity in comparison with the temperature. For linear alkanes,

c decreases (13 for n-hexane and 6.3 for octadecane) with the mo-lar mass [60]. Our result for 2-alkylamines suggests that, as for al-

kanes, when the number of carbon atoms increases the molecular

flexibility also increases, and that softens the intermolecular po-

tential, namely, the value ofc decreases.In our previous work [1], where we have estimated the values of

c for three 1-alkylamines in the same way, the same tendency of c

to decrease with the increase of carbon atoms has been observed.

From the viewpoint of the difference of molecular structure, it is

worse to compare the value ofc among the structural isomers of1-alkylamines and 2-alklamines. 2-alkylamines that have branch-

ing molecular structure show lower scaling factor than 1-alkyl-

amines that have linear molecular structure, namely,

pentylamine (c = 8.17) versus 2-aminopentane (c = 7.77) and hep-tylamine (c = 7.53) versus 2-aminoheptane (c = 7.24). These differ-

ences ofc suggest that, in the comparison of molecules that havethe same number of carbon atoms, molecules with the branching

structure have a lower intermolecular potential than the ones with

the linear structure.

5. Conclusion

A total of 93 experimental dynamic viscosity measurements are

reported for 2-aminobutane, 2-aminopentane, 2-aminoheptane

and 2-aminooctane, for temperatures between 293.15 K and353.15 K and up to 100 MPa. At atmospheric pressure

(p = 0.1 MPa) the dynamic viscosity was measured by a classical

capillary viscometer (Ubbelohde) with an experimental uncer-

tainty of 1%, whereas the viscosity under pressure was measured

with a falling-body viscometer with an experimental uncertainty

of 2%.

The experimental data for these systems have been used in or-

der to evaluate the performance of one empirical correlation (Vo-

gelFulcherTamman representation combined with Tait-like

equation; seven adjustable parameters for each compound) as well

as six models with a more or less developed physical and theoret-

ical background: the rough hard-sphere scheme (five adjustable

parameters), a viscosity model based on the free-volume concept

(three adjustable parameters), the f-theory model based on friction

consideration (two adjustable parameters), a correlation derived

from molecular dynamics (one adjustable parameter), a model

based on Eyring theory combined with cubic EoS (3 adjustable

parameters) and a viscosity representation based on thermody-

namic scaling (one adjustable parameter). This evaluation shows

that some simple models can represent the viscosity of these sys-

tems within an acceptable and satisfactory uncertainty. Further, by

performing the linearization of characteristic parameters on molar

mass, and by comparing with the results of 1-alkylamines obtained

in previous work, the way different models take into account the

effect of molecular structure (carbon chain length and position of

amino group) has been revealed.

The free-volume model, the f-theory model, the molecular

dynamics viscosity model and the Eyring-PR EoS model are all

applicable to gases, liquids, and dense fluids. Because of this, these

four models are suitable for industrial processes involving different

phases or phase changes. Moreover, from a fundamental point of

view, the rough hard-sphere scheme, the free-volume model, the

molecular dynamics model, and Eyring-PR EoS model provide

some insight on the microstructure of these complex systems. Fi-

nally, it should be underlined that the four 2-alkylamines consid-

ered here and also the three 1-alkylamines reported in previous

work [1] show a pressure and temperature viscosity behaviour

compatible with the superposition and scaling viscosity scheme,

providing one single master curve.

References

[1] M. Yoshimura, C. Boned, A. Baylaucq, G. Galliro, H. Ushiki, J. Chem.Thermodyn. 41 (2009) 291.

Fig. 8. Viscosity versus qc/T(g =f(TVc)) (h: 2-aminobutane,}: 2-aminopentane,4:2-aminoheptane and s: 2-aminooctane).



12/12

[2] M. Yoshimura, A. Baylaucq, J.-P. Bazile, H. Ushiki, C. Boned, J. Chem. Eng. Data

54 (2009) 1702.

[3] H. Vogel, Phys. Z. Leipzig 22 (1921) 645.

[4] G.S. Fulcher, J. Am. Ceram. Soc. 8 (1925) 339.

[5] G. Tamman, W. Hesse, Z. Anorg. Allg. Chem. 156 (1926) 245.

[6] J.H. Dymond, M.A. Awan, Int. J. Thermophys. 10 (1989) 941.

[7] M.J. Assael, J.H. Dymond, M. Papadaki, P.M. Patterson, Fluid Phase Equilibr. 75

(1992) 245.

[8] A. Allal, C. Boned, A. Baylaucq, Phys. Rev. E 64 (2001) 011203.

[9] A. Allal, M. Moha-Ouchane, C. Boned, Phys. Chem. Liq. 39 (2001) 1.

[10] C. Boned, A. Allal, A. Baylaucq, C.K. Zberg-Mikkelsen, D. Bessieres, S.E.Quiones-Cisneros, Phys. Rev. E 69 (2004) 031203.

[11] S.E. Quiones-Cisneros, C.K. Zeberg-Mikkelsen, E.H. Stenby, Fluid Phase

Equilibr. 169 (2000) 249.

[12] S.E. Quiones-Cisneros, C.K. Zeberg-Mikkelsen, E.H. Stenby, Fluid Phase

Equilibr. 178 (2001) 1.

[13] G. Galliro, C. Boned, A. Baylaucq, Ind. Eng. Chem. Res. 44 (2005) 6963.

[14] R. Macas-Salinas, F. Garca-Snchez, O. Hernndez-Garduza, AIChE J. 49

(2003) 799.

[15] G. Cruz-Reyes, G. Luna-Brcenas, J.F.J. Alvarado, I.C. Snchez, R. Macas-Salinas,

Ind. Eng. Chem. Res. 44 (2005) 1960.

[16] C.M. Roland, S. Hensel-Bielowka, M. Paluch, R. Casalini, Rep. Progr. Phys. 68

(2005) 1405.

[17] C.M. Roland, S. Bair, R. Casalini, J. Chem. Phys. 125 (2006). 124508.

[18] P. Daug, A. Baylaucq, L. Marlin, C. Boned, J. Chem. Eng. Data 46 (2001) 823.

[19] C. Zberg-Mikkelsen, A. Baylaucq, G. Watson, C. Boned, Int. J. Thermophys. 26

(2005) 1289.

[20] A.S. Pensado, M.J.P. Comuas, L. Lugo, J. Fernandez, J. Chem. Eng. Data 50

(2005) 849.

[21] M.C.S. Chen, J.A. Lescarboura, G.W. Swift, AIChE J. 14 (1968) 123.

[22] Y.L. Sen, E. Kiran, J. Supercrit. Fluids 3 (1990) 91.

[23] E. Kiran, Y.L. Sen, Int. J. Thermophys. 13 (1992) 411.

[24] M.J. Assael, H.M.T. Avelino, N.K. Dalaouti, J.M.N.A. Fareleira, K.R. Harris, Int. J .

Thermophys. 22 (2001) 789.

[25] M.L. Huber, A. Laesecke, H.W. Xiang, Fluid Phase Equilibr. 224 (2004) 263.

[26] I. Cibulka, L. Hnedkovsky, J. Chem. Eng. Data 41 (1996) 657.

[27] H.W. Xiang, A. Laesecke, M.L. Huber, J. Phys. Chem. Ref. Data 35 (2006) 1597.

[28] P.S. Rao, K.C. Rao, G.N. Swamy, Int. J. Chem. Sci. 4 (2006) 553.

[29] M.E. Bai, K.G. Neerajakshi, K.S.V.K. Rao, G.N. Swamy, M.C.S. Subha, J. Indian

Chem. Soc. 82 (2005) 25.

[30] M.C.S. Subha, G.N. Swamy, M.E. Bai, K.S.V.K. Rao, Phys. Theor. Anal. Chem. 43

(2004) 1876.

[31] M.A. Saleh, S. Akhtar, A.R. Khan, Phys. Chem. Liq.: Int. J. 39 (2001) 85.

[32] L.I. Kioupis, E.J. Maginn, J. Phys. Chem. B 104 (2000) 7774.

[33] M.J.P. Comuas, A. Baylaucq, C. Boned, J. Fernndez, Int. J. Thermophys. 22

(2001) 749.

[34] E.N. Andrade, Philos. Mag. 17 (1934) 497.

[35] J. Millat, J.H. Dymond, C.A. Nieto de Castro, Transport Properties of Fluids,

Cambridge University Press, UK, 1996.

[36] A. Baylaucq, C. Boned, P. Dauge, P. Xans, Phys. Chem. Liq. 37 (1999) 579.

[37] A. Baylaucq, M. Moha-ouchane, C. Boned, Phys. Chem. Liq. 38 (2000) 353.

[38] M.J.P. Comuas, A. Baylaucq, F. Plantier, C. Boned, J. Fernndez, Fluid Phase

Equilibr. 222223 (2004) 331.

[39] P. Reghem, A. Baylaucq, M.J.P. Comuas, J. Fernandez, C. Boned, Fluid PhaseEquilibr. 236 (2005) 229.

[40] T.H. Chung, M. Ajlan, L.L. Lee, K.E. Starling, Ind. Eng. Chem. Res. 27 (1988) 671.

[41] S. Chapman, T.G. Cowling, D. Burnett, The Mathematical Theory of Non-

Uniform Gases, Cambridge University Press, UK, 1970.

[42] P.D. Neufeld, A.R. Janzen, R.A. Aziz, J. Chem. Phys. 57 (1972) 1100.

[43] K.N. Marsh, C.L. Young, D.W. Morton, D. Ambrose, C.Tsonopoulos, J. Chem. Eng.

Data 51 (2006) 305.

[44] B.E. Poling, J.M. Prausnitz, J.P. OConnell, The Properties of Gases and Liquids.

The Properties of Gases and Liquids, McGraw-Hill, New York, USA, 2001.

[45] Y. Nannoolal, J. Rarey, D. Ramjugernath, Fluid Phase Equilibr. 252 (2007) 1.

[46] Q. Wang, P. Ma, Q. Jia, S. Xia, J. Chem. Eng. Data 53 (2008) 1103.

[47] Q. Wang, Q. Jia, P. Ma, J. Chem. Eng. Data 53 (2008) 1877.

[48] Q. Jia, Q. Wang, P. Ma, J. Chem. Eng. Data 53 (2008) 2606.

[49] D. Ambrose, J. Walton, Pure Appl. Chem. 61 (1989) 1395.

[50] S.P. Tan, H. Adidharma, B.F. Towler, M. Radosz, Ind. Eng. Chem. Res. 44 (2005)

8409.

[51] D.-Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fund. 15 (1976) 59.

[52] G. Soave, Chem. Eng. Sci. 27 (1972) 1197.

[53] C.K. Zberg-Mikkelsen, A. Baylaucq, M. Barrouhou, S.E. Quiones-Cisneros, C.

Boned, Fluid Phase Equilibr. 222223 (2004) 135.

[54] S.P. Tan, H. Adidharma, B.F. Towler, M. Radosz, Ind. Eng. Chem. Res. 45 (2006)

2116.

[55] S.E. Quiones-Cisneros, U.K. Deiters, J. Phys. Chem. B 110 (2006) 12820.

[56] S. Glasstone, K.J. Laidler, H. Eyring, The Theory of Rate Processes: The Kinetics

of Chemical Reactions Viscosity Diffusion and Electrochemical Phenomena,

McGraw-Hill, New York, 1941.

[57] Q.-F. Lei, Y.-C. Hou, R.-S. Lin, Fluid Phase Equilibr. 140 (1997) 221.

[58] D. Coslovich, C.M. Roland, J. Phys. Chem. B 112 (2008) 1329.

[59] O. Fandino, M.J.P. Comuas, L. Lugo, E.R. Lopez, J. Fernandez, J. Chem. Eng. Data

52 (2007) 1429.

[60] A.S. Pensado, A.A.H. Pdua, M.J.P. Comuas, J. Fernndez, J. Phys. Chem. B 112

(2008) 5563.


Documents

Chemical Physics 369 (2010) 126–137