ORIGINAL PAPER

Nonlinear modeling of equilibrium sorption of selectedanionic adsorbates from aqueous solutions on cellulosicsubstrates. Part 1: model development

Changhai Xu • Renzo Shamey

Received: 30 July 2011 / Accepted: 15 February 2012 / Published online: 2 March 2012

� Springer Science+Business Media B.V. 2012

Abstract A new nonlinear isothermal sorption

model, incorporating Donnan equilibrium and electri-

cal neutrality in the classical sorption model of direct

dyes onto cellulosic substrates, as model adsorbates, is

proposed. The nonlinear isothermal model was used to

simulate equilibrium sorption of adsorbates containing

ionic charges (z) of -2 to -4 on cellulose adsorbents

at various temperature (T) and sodium chloride

concentrations ([NaCl]). A detailed analysis of simu-

lation results demonstrates that results based on the

nonlinear sorption model highly agree with those

based on the log-linear sorption model when the

deviation in the concentration of sodium ions in the

aqueous solution ([Na?]S) relative to [NaCl] used in

the sorption system is restricted to\5.0%. Compared

to the log-linear model, the nonlinear model avoids

using graphical techniques that are relatively insensi-

tive for determining important sorption parameters

such as the internal accessible volume (V) and the

standard affinity associated with sorption (-Dl�). The

nonlinear sorption model was used to examine the

correlation of fit for previously reported sorption data.

The model parameters V and -Dl� based on curve fits

were used to estimate V for cellulose as well as -Dl�.

The values were found to match those based on the

conventional log-linear model when deviations of

[Na?]S relative to ½Naþ�S were below 5%. The

nonlinear model therefore provides a convenient and

accurate technique to interpret the sorption of a range

of anionic adsorbates on cellulosic substrates.

Keywords Nonlinear modeling and analysis �Equilibrium sorption � Sorption isotherm � Anionic

adsorbate � Cellulose � Internal accessible volume

Introduction

Sorption of adsorbates from aqueous solutions on

substrates is a complex process which is generally

controlled by adsorbate–substrate interactions involv-

ing various types of intermolecular forces, such as

electrostatic (ionic) (Sumner 1986; McGregor and

Iijima 1992), van der Waals (London) (Bird et al.

2006), polar (hydrogen bonding) (Yamaki et al. 2005),

and hydrophobic (cooperative binding) interactions

(Nemethy and Scheraga 1962; Maruthamuthu and

Sobhana 1979; Yang 1993). The interaction forces

involved in a sorption system depend on the physical

and chemical properties of adsorbates and adsorbents.

Anionic dyes can be considered a model class of

anionic adsorbates applied to cellulosic substrates.

Direct dyes were among the first classes whose

sorption properties on cellulosic substrates were

examined (Vickerstaff 1954). They are anionic mol-

ecules with linear and coplanar structures and possess

C. Xu � R. Shamey (&)

Fiber and Polymer Science Program, North Carolina State

University, Raleigh, NC 27695-8301, USA

e-mail: rshamey@ncsu.edu

123

Cellulose (2012) 19:615–625

DOI 10.1007/s10570-012-9675-7

inherent substantivity to cellulosic substrates. In an

aqueous solution, cellulose acquires negative charges

on its surface which repels the direct dye anions.

Electrolytes are usually added into the dyebath in

part to provide sodium counter-ions to minimize

the effect of the negative charge on the surface of

cellulose. This facilitates the adsorption of dye anions

by the substrate. A fundamental examination of the

sorption of direct dye on cellulose can elucidate a

better understanding of the sorption behavior of

anionic adsorbates on cellulosic substrates.

In this paper, a nonlinear sorption model is

developed that utilizes the classical sorption isotherm

of direct dyes from an aqueous solution on cellulosic

substrates. The nonlinear sorption model was used to

simulate equilibrium sorption of adsorbates with ionic

charge of -2 to -4 on cellulose adsorbents under

various conditions. In addition, the nonlinear sorption

model was fitted to isothermal sorption data previ-

ously reported (Hanson et al. 1935; Willis et al. 1945;

Marshall and Peters 1947; Peters and Vickerstaff

1948) and the performance of the nonlinear sorption

model was evaluated by comparing to the log-linear

sorption model.

Classical sorption isothermal models

The sorption behavior of direct dyes on cellulose

substrates is classically described by a two-phase

sorption model developed in the first half of the

twentieth century (Hanson et al. 1935; Willis et al.

1945; Marshall and Peters 1947; Peters and Vicker-

staff 1948). This sorption model assumes two phases,

the internal cellulose phase (F) and the external

aqueous solution phase (S), and that any adsorbed ion

on cellulose substrate is dissolved in the internal

phase. Figure 1 shows the distribution of ions at

equilibrium in the two-phase sorption system contain-

ing a direct dye (NazD) and sodium chloride (NaCl).

Assuming that both NazD and NaCl completely

dissociate in the sorption system, the equilibrium

sorption is interpreted using Eq. 1,

�Dl� ¼ RT ln½Dz��F½Naþ�zF

Vzþ1½Dz��S½Naþ�zS

� �ð1Þ

where [Dz-]F and [Na?]F are the concentrations of dye

anions and sodium ions on cellulose (mol kg-1);

[Dz-]S and [Na?]S are the concentrations of dye anions

and sodium ions in the aqueous solution (mol L-1);

z is the ionic charge on dye; V is the internal accessible

volume of cellulose substrate (L kg-1); R is the gas

constant (J K-1); and T is temperature (K).

-Dl� is defined as the change in standard chemical

potential of the direct dye from the aqueous solution to

the substrate and is commonly called the affinity of a

direct dye toward a particular cellulose substrate.

-Dl� is regarded as the driving force in the kinetics

of the sorption process. It is thus essential to calculate

-Dl� of a sorption system to understand the behavior

of the adsorbate on the substrate.

Fig. 1 Distribution of ions in the two-phase equilibrium

sorption system

Fig. 2 Simulated sorption isotherm of an anionic adsorbate

with a charge of -2 on a cellulosic substrate with an internal

accessible volume of 0.25 L kg-1, in the presence of

0.025 mol L-1 NaCl at 40 �C

616 Cellulose (2012) 19:615–625

123

In Eq. 1, with the exception of V, all the model

parameters can be directly measured or indirectly

calculated. [Dz-]S is commonly measured in the

residual dyebath via spectrophotometric or colorimet-

ric methods and [Dz-]F can be determined either from

the calculation of the depletion of dye in the aqueous

solution or by extraction of dye from the substrate.

[Na?]S is considered to be equivalent to the sum of all

counter ions in the dyebath including the dye and

chloride anions, as shown in Eq. 2, providing the

volume of the dyebath is in a large excess of the

internal accessible volume of cellulose substrate.

½Naþ�S ¼ z½Dz��S þ ½Cl��S ð2Þ

[Na?]F is generally calculated from Eq. 3 which is

derived from the Donnan equilibrium between fiber

and aqueous solution (Eq. 4) and the electrical neu-

trality condition of the cellulose substrate (Eq. 5).

-21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8-19

-18

-17

-16

-15

-14

-13

-12

-11

-10

-9

-8

r = 0.9917Slope = 0.7894

Simulated sorption data Linear fit

ln([

D2-] F

[Na+

]2 F)

ln([D2-]S[Na+]2

S)

Fig. 3 Linear fit of simulated sorption data to the log-linear

sorption model

Fig. 4 Deviations in [Na?]S relative to [NaCl] for the simulated

sorption data

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.00070.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

[D2-] F

, mol

kg-1

[D2-]S, mol L-1

(a)

(b)

(c)

-21 -20 -19 -18 -17 -16 -15 -14-20

-19

-18

-17

-16

-15

-14

-13

-12

r = 0.9999Slope = 0.9860

Simulated sorption data Linear fit

ln([

D2-] F

[Na+

]2 F)

ln([D2-]S[Na+]2

S)

Fig. 5 Achievement of better agreement between a the non-

linear simulation and b the linear fit of simulated sorption data

by c reducing deviations in [Na?] relative to [NaCl] to less

than 5%

Cellulose (2012) 19:615–625 617

123

½Naþ�F ¼

0:5 z½Dz��F þ z2½Dz��2F þ 4V2½Naþ�S½Cl��S� �0:5

� �

ð3Þ

½Naþ�F½Cl��FV2

¼ ½Naþ�S½Cl��S ð4Þ

½Naþ�F ¼ z½Dz��F þ ½Cl��F ð5Þ

The model parameter V is believed to be a

function of the physical structure of the cellulosic

substrate and may vary from one type of cellulosic

substrate to the other. V is commonly used to

compare and contrast the difference in physical

structures of cellulosic substrates (Standing 1954;

Holmes 1958; Carrillo et al. 2002; Ibbett et al. 2006,

2007). As implied in Eq. 1, V can greatly influence

-Dl� and must therefore be assigned a precise

value, especially when comparing sorption behaviors

of the same adsorbate on a variety of cellulosic

substrates. In order to determine the parameter V,

Eq. 1 is usually expanded in a log-linear model as

given in Eq. 6.

ln ½Dz��F½Naþ�zF�

¼ ln ½Dz��S½Naþ�zS� þ ðzþ 1Þ lnðVÞ þ �Dl�

RTð6Þ

By adjusting the V value a linear relationship

between ln([Dz-]F[Na?]Fz ) and ln([Dz-]S[Na?]S

z) can

be obtained. However, care should be exercised to

ensure that not only the best-fit straight line has a

slope of unity but also �Dl� calculated from Eq. 1

remains constant for all sorption data. The best fit of

Eq. 6 to sorption data is routinely obtained by

assigning an arbitrary value to V by the trial-and-error

method, and often involves tedious mathematical

calculations. The log–log plot of sorption data,

however, is not very sensitive to changes in the value

assigned to V. This can be improved by the use of a

nonlinear modeling approach which is described in the

following section.

Development of a nonlinear isothermal model

Equation 1 is rearranged to give Eq. 7.

½Dz��S ¼½Naþ�zF

½Naþ�zSVzþ1 exp �Dl�

RT

� � ½Dz��F ð7Þ

Equation 3 is then substituted into Eq. 7 leading to

Eq. 8.

Equation 8 is a nonlinear isothermal sorption

model expressing [Dz-]S in terms of [Dz-]F and can

be used to characterize the sorption of anionic

adsorbates on cellulose. In Eq. 8, [Cl-]S, z, R and T

are known parameters, and [Na?]S can be calculated

from Eq. 2. Providing [Na?]S is relatively constant for

a specific sorption isotherm, V and -Dl� can be

obtained by fitting Eq. 8 to sorption data.

Simulation of isothermal adsorption

It is assumed that an anionic adsorbate is adsorbed

onto a cellulosic substrate in aqueous solution and the

sorption process is allowed to reach equilibrium.

Accordingly, the sorption isotherm can be simulated

using the nonlinear model (Eq. 8). Figure 2 shows an

example of nonlinear simulation by which the sorption

isotherm of an adsorbate with a charge of -2 towards a

cellulosic substrate. The internal accessible volume of

cellulose is considered to be 0.25 L kg-1 in the

presence of 0.025 mol L-1 NaCl at 40 �C in the

adsorbate concentration ([D2-]F) range of 0–

0.025 mol kg-1 with 50 evenly spaced data points.

All the sorption data based on the nonlinear simulation

should follow a strictly linear relationship with a slope

of unity, as given by Eq. 6, when they are plotted in

log–log form, and the correlation coefficient (r) should

be 1 providing the two models are equivalent in

describing the hypothesized sorption system. How-

ever, Fig. 3 shows that the linear fit of simulated

sorption data against the log-linear sorption model

½Dz��S ¼½Dz��F z½Dz��F þ z2½Dz��2F þ 4V2½Naþ�S½Cl��S

� �0:5� �z

2z½Naþ�zSVzþ1 exp �Dl�

RT

� � ð8Þ

618 Cellulose (2012) 19:615–625

123

produces a straight line with a slope deviating from

unity in spite of the relatively high correlation

coefficient obtained. This disagreement is most likely

caused by the parameter [Na?]S that is assumed to be

constant in the nonlinear simulation (here approxi-

mated by 0.025 mol L-1 NaCl). However, [Na?]S is a

function of [D2-]S according to Eq. 2. Figure 4 shows

the % deviations associated with [Na?]S relative to

[NaCl] (Eq. 9) based on simulated concentration of

adsorbate anions in the solution ([D2-]S). [D2-]S

values used in Fig. 4 were in turn derived from the

concentration of adsorbate anions in the fiber on a

range of 0–0.025 mol kg-1 using Eq. 8. Results show

that an increase in [D2-]S results in an increased

deviation in [Na?]S. For the nonlinear simulations in

the [D2-]F range of 0–0.007 mol kg-1 (Fig. 5a),

errors of [Na?]S relative to [NaCl] for all 50 data

points are below 5% (Fig. 5b). Here the linear fit of

simulated sorption data produces a straight line with a

slope closely approximating unity and a high corre-

lation coefficient (Fig. 5c). Table 1 summarizes non-

linear simulations of hypothesized isothermal sorption

data in the [Dz-]F range of 0–0.025 mol kg-1 based

on a linear fit as well as the adjusted linear fit values

when the deviation in [Na?]S is kept below 5%. The

adjusted values generate a high level of agreement

between the nonlinear sorption model and the log-

linear sorption model in describing the sorption

behavior of an anionic adsorbate on a cellulosic

substrate.

Table 1 Agreement of the nonlinear sorption model with the log–log sorption model

Parameters for nonlinear simulationa Linear fitb Adjusted linear fitc

z [NaCl] (mol L-1) T ( �C) V (L kg-1) -Dl� (J mol-1) r Slope [Dz-]F (mol L-1) r Slope

2 0.025 40 0.25 15,000 0.9917 0.7894 0–0.007 0.9999 0.9860

3 0.025 40 0.25 20,000 0.9683 0.5633 0–0.006 0.9999 0.9863

4 0.025 40 0.25 25,000 0.9562 0.4025 0–0.005 0.9999 0.9898

2 0.050 40 0.25 15,000 0.9994 0.9466 0–0.014 0.9999 0.9859

3 0.050 40 0.25 20,000 0.9953 0.8719 0–0.012 0.9999 0.9863

4 0.050 40 0.25 25,000 0.9778 0.7183 0–0.011 0.9999 0.9851

2 0.025 80 0.25 15,000 0.9872 0.7092 0–0.005 0.9999 0.9874

3 0.025 80 0.25 20,000 0.9683 0.4863 0–0.004 0.9999 0.9902

4 0.025 80 0.25 25,000 0.9709 0.3708 0–0.004 0.9999 0.9872

2 0.050 80 0.25 15,000 0.9982 0.9067 0–0.010 0.9999 0.9875

3 0.050 80 0.25 20,000 0.9878 0.7689 0–0.009 0.9999 0.9863

4 0.050 80 0.25 25,000 0.9631 0.5707 0–0.008 0.9999 0.9872

2 0.025 40 0.5 15,000 0.9994 0.9465 0–0.014 0.9999 0.9860

3 0.025 40 0.5 20,000 0.9953 0.8719 0–0.012 0.9999 0.9863

4 0.025 40 0.5 25,000 0.9778 0.7183 0–0.011 0.9999 0.9851

2 0.050 40 0.5 15,000 0.9999 0.9892 0–0.025 0.9999 0.9892

3 0.050 40 0.5 20,000 0.9999 0.9844 0–0.025 0.9999 0.9844

4 0.050 40 0.5 25,000 0.9997 0.9751 0–0.022 0.9998 0.9851

2 0.025 80 0.5 15,000 0.9982 0.9067 0–0.010 0.9999 0.9874

3 0.025 80 0.5 20,000 0.9878 0.7689 0–0.009 0.9999 0.9863

4 0.025 80 0.5 25,000 0.9631 0.5707 0–0.008 0.9999 0.9872

2 0.050 80 0.5 15,000 0.9999 0.9797 0–0.021 0.9999 0.9862

3 0.050 80 0.5 20,000 0.9997 0.9642 0–0.019 0.9999 0.9840

4 0.050 80 0.5 25,000 0.9983 0.9310 0–0.017 0.9999 0.9840

a Hypothesized valuesb In the [D2-]F range of 0–0.025 mol L-1

c After reducing deviation in [Na?]S relative to [NaCl] below 5%

Cellulose (2012) 19:615–625 619

123

% Deviation in ½Naþ�s relative to [NaCl]

¼ z½Dz��½NaCl� � 100 ð9Þ

Model validation

Previously published experimental sorption data of

direct dyes from aqueous solution on cellulosic

substrates (Hanson et al. 1935; Willis et al. 1945;

Marshall and Peters 1947; Peters and Vickerstaff

1948) were employed to evaluate the performance

of the proposed nonlinear sorption model. The

structures of direct dyes used in these investigations

are shown in Fig. 6. Table 2 summarizes the

reported experimental data including the direct dyes

and the cellulosic substrates employed, as well as

the dyeing conditions. The full equilibrium sorption

data in each dataset are not given for sake of

brevity. The average concentration of sodium ions

ð½Naþ�sÞ of all data points for each dataset was also

included in Table 2 to approximate [Na?]S in the

nonlinear sorption model.

H

H

NaO3S

SO3Na

NN N OC2H5

NC2H5O

CI Direct Yellow 12

C

O

NN

O

H H

CI Direct Red 24

CI Direct Blue 1

O

N

SO3Na

N

NaO3S

NN

H H

H3CONaO3S CH3

NN

OCH3

H3CO

H

NH

N O

O

H2N

NH2

NaO3S

NaO3S

SO3Na

SO3Na

Fig. 6 Structures of dyes

used in previous studies

Table 2 Previously published experimental data

Dataset Dyes used z Cellulosic substrate T (�C) [NaCl] (mol L-1) ½Naþ�S (mol L-1)

1a CI direct yellow 12 2 Cellophane sheet 60 0.0086 0.0094

2a CI direct yellow 12 2 Cellophane sheet 60 0.0684 0.0691

3b CI direct red 24 3 Viscose rayon 60 0.0100 0.0109

4b CI direct red 24 3 Viscose rayon 90 0.0200 0.0213

5c CI direct blue 1 4 Cotton fiber 90 0.0856 0.0861

a See Willis et al. (1945), Peters and Vickerstaff (1948)b See Marshall and Peters (1947)c See Hanson et al. (1935), Peters and Vickerstaff (1948)

620 Cellulose (2012) 19:615–625

123

The nonlinear sorption model was fitted to data

shown in Table 2. All curve fitting was performed

using Origin 7.5 (OriginLab). The best-fit curves are

shown in Fig. 7. The goodness of fit was evaluated by

the coefficient of determination (R2) as given in

Eq. 10.

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012

(a)

R2 = 0.9869

D2-

S, mol L-1

D2-

F, m

ol k

g-1

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

R2 = 0.9981

D2-

S, mol L-1

D2-

F, m

ol k

g-1

(b)

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

R2 = 0.9839

D3-

S, mol L-1

D3-

F, m

ol k

g-1

(c)

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012

R2 = 0.9905

D3-

S, mol L-1

D3-

F, m

ol k

g-1

(d)

0.000

0.002

0.004

0.006

0.008

0.010

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005

R2 = 0.9757

D4-

S, mol L-1

D4-

F, m

ol k

g-1

(e)

Fig. 7 Best-fit curves of reported experimental data shown in Table 2 using the nonlinear sorption model: a Dataset 1, b Dataset 2,

c Dataset 3, d Dataset 4 and e Dataset 5

Cellulose (2012) 19:615–625 621

123

R2 ¼ 1�P½Dz��S � ½Dz��Fit

S

� 2

P½Dz��S � ½Dz��S� �2

ð10Þ

where [Dz-]SFit is the value obtained from the nonlinear

curve fitting corresponding to the observed value of

[Dz-]S, and ½Dz��S is the mean of all [Dz-]S values. The

best-fit values of parameters V and -Dl� as well as

their reported values are summarized in Table 3.

The V values were directly obtained from published

manuscripts and are all based on the log-linear model.

Table 3 shows that the V and -Dl� values determined

based on the nonlinear sorption model roughly match

those reported based on the log-linear sorption model.

The exception is the case of Dataset 3 which contains

some data points with [Na?]S deviations relative to

½Naþ�S (based on Eq. 11) greater than 5%. As men-

tioned, the model parameter [Na?]S in Eq. 8 is

assumed to be constant for all sorption data employed

in an isotherm. In a real sorption system, however,

[Na?]S depends on the concentration of sodium

chloride and the equilibrium concentration of anionic

adsorbate in the aqueous system (Eq. 2). A scatter plot

of [Na?]S deviations relative to ½Naþ�S against [D3-]S

for Dataset 3 is shown in Fig. 8a which demonstrates

the deviations associated with data points 1, 2 and 7

exceed the 5% threshold. Equations 2 and 9 indicate

that these deviations are attributed to too high [D3-]S

compared to [NaCl] used in the sorption system. When

the large deviations at high [D3-]S values (i.e. points 5,

6 and 7 shown in Fig. 8a) are removed from Dataset 3,

the deviations of [Na?]S relative to ½Naþ�S for each

data point are reduced below 5% (Fig. 8b). The

nonlinear sorption model can then be fitted to the

adjusted dataset. The best-fit curve for the adjusted

Dataset 3 is given in Fig. 8c. The nonlinear curve

fitting results in a V value of 0.64 ± 0.12 L kg-1 and a

-Dl� value of 15.6 ± 1.6 kJ mol-1 which reason-

ably approximate the reported values shown in

Table 3.

% Deviation in ½Naþ�s relative to [Naþ�s

¼ ½Naþ�s � ½Naþ�s½Naþ�s

� 100 ð11Þ

Figure 8 shows that the model functions well over a

wide range of concentrations providing the few data

points that exhibit high levels of deviation are

removed from the dataset.

Nonlinear versus linear Modeling

Figure 9 shows an example, for Dataset 5, of graphic

techniques that are essentially used in the log-linear

sorption models to determine the value of V for

cellulosic adsorbent. The graph also demonstrates the

advantage of the nonlinear sorption model over the log-

linear sorption model, in describing the sorption behav-

ior of anionic adsorbates on cellulosic adsorbents.

Figure 9a shows that the logarithmic form of the

sorption data, i.e. ln([D4-]F[Na?]F4) versus ln([D4-]S

[Na?]S4) can exhibit various correlations when a random

value of V is applied to [Na?]F (Eq. 6). The plots of

slope and correlation coefficient resulting from linear

fitting against V are shown in Fig. 9b. It can be seen that

when the value of V increases from 0.05 to 0.5 L kg-1

Table 3 Values of V and -Dl� resulting from nonlinear fitting in comparison with the reported values

Dataset Nonlinear fitting Reported linear fitting

V (L kg-1) -Dl� (kJ mol-1) V (L kg-1) -Dl� (kJ mol-1)

1 0.64 ± 0.28 14.8 ± 3.3 0.45 17.1

2 0.50 ± 0.06 17.7 ± 0.8 0.45 18.2

3 0.0003 ±a 98.4 ±a 0.45 18.1

3b 0.64 ± 0.12 15.6 ± 1.6 0.45 18.1

4 0.58 ± 0.11 14.1 ± 1.6 0.45 15.9

5 0.16 ± 0.02 27.9 ± 1.4 0.22 24.4

a Error out of calculation range

b Adjusted dataset 3 excluding points with [Na?]S deviations relative to ½Naþ�S [ 5%

622 Cellulose (2012) 19:615–625

123

the slopes of straight lines vary in a wide range in spite

of associated high correlation coefficients (0.995–1).

Therefore, the determination of the best-fit curve in

terms of the correlation coefficient of the linear fit can be

challenging. If the log-linear sorption model is forced

to fit sorption data at a slope of unity, a V value of

0.19 L kg-1 with a correlation coefficient of 0.9991

would be obtained. This V value is close to the value

based on the nonlinear sorption model (0.16 L kg-1)

as well as the reported value (0.22 L kg-1). The

corresponding value of -Dl� would be 25.7 versus

27.9 kJ mol-1 based on the nonlinear sorption model

and the reported value of 24.4 kJ mol-1. Hence, the

nonlinear sorption model is roughly equivalent to the

log-linear sorption model in describing the sorption

behavior of anionic adsorbates on cellulosic adsorbents

while avoiding the relatively insensitive graphical

techniques that are required to determine model

parameters in the log-linear sorption model.

Conclusions

A nonlinear isothermal sorption model incorporating

Donnan equilibrium and electrical neutrality was

developed to describe the equilibrium sorption of

0.000

0.001

0.002

0.003

0.004

0.005

0.0000 0.0001 0.0002 0.0003

R2 = 0.9941V = 0.64 0.12 L kg-1

−Δμo = 15.6 1.6 kJ mol-1

D3-

S, mol L-1

D3-

F, m

ol k

g-1

(c)

(b)

(a)

Fig. 8 Improved correlation of the experimental data with the

nonlinear sorption model: a original dataset, b adjusted dataset

containing points with [Na?]S deviations \5%, and c best-fit

curve of the adjusted dataset

-22 -21 -20 -19 -18 -17-24

-23

-22

-21

-20

-19

-18

-17

-16

-15

V = 0.1 L kg-1

V = 0.2 L kg-1

V = 0.4 L kg-1

ln([

D4-] F

[Na+

]4 F)

ln([D4-]S[Na+]4

S)

(a)

0.0 0.1 0.2 0.3 0.4 0.5 0.60.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6(b)

V, L kg -1

Slo

pe

Slope = 1V = 0.19 L kg-1

r = 0.9991

0.995

0.996

0.997

0.998

0.999

1.000

Correlation coefficient (r)

Fig. 9 An example of graphical techniques: a the log-linear

sorption model fitting to Dataset 5 using a random value of V and

b plots of slope and correlation coefficient against V

Cellulose (2012) 19:615–625 623

123

anionic adsorbates, using direct dyes as model com-

pounds, from aqueous solutions on cellulosic sub-

strates. The model expresses the equilibrium

concentration of an anionic adsorbate in aqueous

solution, [Dz-]S, in terms of the equilibrium concen-

tration of the adsorbate on cellulose, [Dz-]F. Six

parameters are included in the nonlinear model: the

ionic charge on adsorbate (z), the internal accessible

volume of cellulose (V), the concentration of sodium

ions in aqueous solution ([Na?]S), the concentration of

chloride ions in aqueous solution ([Cl-]S), tempera-

ture of the isothermal sorption system (T), and the

affinity of adsorbate for cellulose (-Dl�). The model

variables [Dz-]S and [Dz-]F and the parameters z,

[Na?]S, [Cl-]S and T can be directly measured or

indirectly calculated. The parameters V and -Dl� can

be obtained by curve fitting of sorption data using the

nonlinear sorption model. The nonlinear sorption

model is thus more convenient and robust compared

to the traditional log-linear model in determining

V and -Dl�.

The nonlinear sorption model was tested by sim-

ulating sorption of adsorbates with ionic charge of -2

to -4 from aqueous solutions on cellulose adsorbents

under various conditions. A detailed analysis of

simulation results showed that the nonlinear sorption

model was equivalent to the log-linear sorption model

in describing the sorption behavior of anionic adsor-

bates on cellulosic substrates if deviations in [Na?]S

for all sorption data are restricted to \5.0%. The

nonlinear sorption model was also fitted to the

previously reported experimental sorption data involv-

ing the application of anionic direct dyes with 2–4

negative charges on various types of cellulosic sub-

strates. It was found that the values of V and -Dl�determined by the nonlinear sorption model roughly

matched the reported values of V and -Dl� for all

datasets except for one that included some data points

with deviations in [Na?]S relative to ½Naþ�S greater

than 5%. The adjustment of the dataset based on

removal of points exhibiting deviations greater than

5% improved the nonlinear curve fitting. The resultant

values of V and -Dl� were found to be close to the

reported values. In comparison to the log-linear

sorption model, the nonlinear sorption model avoids

the uncertainty associated with the use of relatively

insensitive graphical techniques required to determine

some important sorption parameters such as V and

-Dl�. Therefore the nonlinear sorption model is a

more effective and convenient method for describing

the sorption behavior of anionic adsorbates from

aqueous solutions on cellulosic substrates.

References

Bird J, Brough N, Dixon S, Batchelor SN (2006) Understanding

adsorption phenomena: investigation of the dye-cellulose

interaction. J Phys Chem B 110:19557–19561

Carrillo F, Lis MJ, Valldeperas J (2002) Sorption isotherms and

behaviour of direct dyes on lyocell fibres. Dyes Pigm

53:129–136

Hanson J, Neale SM, Stringfellow WA (1935) Absorption of

dyestuffs by cellulose. Part VI. The effect of modification

of the cellulose, and a theory of the electrolyte effect. Trans

Faraday Soc 31:1718–1730

Holmes FH (1958) The absorption of Chrysophenine G by

cotton: a test of quantitative theories of direct dyeing. Trans

Faraday Soc 54(8):1172–1178

Ibbett RN, Phillips DAS, Kaenthong S (2006) Evaluation of a

dye isotherm method for characterisation of the wet-state

structure and properties of lyocell fibre. Dyes Pigm 71:

168–177

Ibbett RN, Phillips DAS, Kaenthong S (2007) A dye-adsorption

and water NMR-relaxation study of the effect of resin

cross-linking on the porosity characteristics of lyocell

solvent-spun cellulosic fibre. Dyes Pigm 75:624–632

Marshall WJ, Peters RH (1947) The heats of reaction and

affinities of direct dyes for cuprammonium rayon, viscose

rayon, and cotton. J Soc Dyers Colour 63:446–461

Maruthamuthu M, Sobhana M (1979) Hydrophobic interactions

in the binding of polyvinylpyrrolidone. J Polym Sci Pol

Chem 17:3159–3167

McGregor R, Iijima T (1992) Dimensionless groups for the

sorption of dye and other ions by polymers. II. Hydro-

chloric acid, C.I. acid blue 25, and polyamides with an

excess of basic groups. J Appl Polym Sci 45:1011–1021

Nemethy G, Scheraga HA (1962) Structure of water and

hydrophobic bonding in protein. III. The thermodynamic

properties of hydrophobic bonds in proteins. J Phys Chem

66:1773–1789

Peters RH, Vickerstaff T (1948) The adsorption of direct dyes on

cellulose. Proc R Soc Lond Ser A 192:292–308

Standing HA (1954) The direct dyeing of cellulose: a method for

the analysis of equilibrium absorption data in terms of

current quantitative theories. J Text Inst 45:T21–T29

Sumner HH (1986) The development of a generalised equation

to determine affinity. Part 1—theoretical derivation and

application to dyeing cellulose with anionic dyes. J Soc

Dyers Colour 102:301–305

Vickerstaff T (ed) (1954) Cellulose-dyeing equilibria with

direct dyes. In: The physical chemistry of dyeing, 2 edn.

Oliver and Boyd, London, pp 191–256

Willis HF, Warwicker JO, Standing HA, Urquhart AR

(1945) The dyeing of cellulose with direct dyes. Part II.

624 Cellulose (2012) 19:615–625

123

The absorption of chrysophenine by cellulose sheet. Trans

Faraday Soc 41:506–541

Yamaki SB, Barros DS, Garcia CM, Socoloski P, Oliveira ON,

Atvars TDZ (2005) Spectroscopic studies of the

intermolecular interactions of Congo Red and tinopal CBS

with modified cellulose fibers. Langmuir 21:5414–5420

Yang Y (1993) Hydrophobic interaction and its effect on cat-

ionic dyeing of acrylic fabric. Text Res J 63:283–289

Cellulose (2012) 19:615–625 625

123