CC BY
92
Section 7. Physics
Shukurlu Y. H.,
doctor of philosophy in physics and mathematics, Sheki Regional Scientific Center of the National Academy of Sciences of the Republic of Azerbaijan,
Sheki, Azerbaijan E-mail: [email protected]; [email protected]
KINETIC PARAMETERS OF THE FIRST AND SECOND STAGES OF THE FISETIN MOLECULE DIFFUSION IN THE FIBROIN FIBER
Abstract: Development of new bioengineering structures made of fibroin - a unique natural silk biopolymer - and its use in regenerative medicine and consumer goods manufacturing is often mentioned in literature data. It is related to the fact that fibroin has many important properties such as biocompatibility, biodegradability, high strength, hygroscopicity and elasticity. All applications of fibroin are related to its physical and chemical properties, one of which is the dyeing affinity of this biopolymer. Dyeing is closely related to the process of mass transfer - diffusion. Consequently, when fibroin is dyed, the most significant part of the dyeing process is the diffusion of dye molecules into microfibrils and this very complex process is not insufficiently studied.
Diffusion is a gradual process, which speed or kinetics is a very important scientific and technological attribute that determines the degree of homogeneity of the adsorbate distribution throughout the adsorbent. We studied the penetration of fisetin molecules into the fibrin microfibrils to determine the effect of different temperatures of the dye solution and the concentration of electrolytes in the solution on the diffusion kinetics parameters.
It is an established fact that the diffusion of the dye continues until reaching an equilibrium concentration in the entire volume of the fiber. Let us mentally divide this process into three stages: 1. adsorption of fisetin molecules on the surface of fibroin fibers; 2. moment when the molecules of fisetin reach the center of the fibroin fiber; 3. begins after the completion of the second and continues until the equilibrium concentration is established in the whole fiber. The first stage occurs almost instantaneously making it impossible to separate this stage from the second stage during the actual dyeing process. Therefore, we shall combine the first and second stages of diffusion.
In this article, by using a three-dimensional physical model of the diffusion of dye in the fiber, we established formulas that describe the kinetics of diffusion of the dye from the adsorption layer to the moment when the fisetin molecules reached the center of the fiber.
Keywords: fisetin dye; molecular diffusion of fisetin; natural silk fibroin; microfibrils of fibroin; temperature of dye solution; concentration of dye molecules in solution; diffusion kinetics parameters; second stage of diffusion; third stage of diffusion.
Introduction
Software is developed to design, calculate and visualize ion-exchange technological schemes. This software aim is to be capable of calculating a multistage sorption process, as well as regeneration and cleaning in several columns of a technological scheme [1]. The relevant literature has many examples of the analytical solution to problems of diffusion kinetics, defined by difficulties in certain geometrical and physical conditions and having no general results that can be used. There is also a second type - those that refuse analytical approach and use scaling and modeling of transfer processes and chemical processes [1]. As suggested by the authors of [2], we also used the third method: the quasistationary method (as named by authors) or an equally accessible surface, due to simplified calculation and detection of physically significant limiting cases.
Silk fibroin is an amphiphilic protein - a chemical compound that has both hydrophilic and hydrophobic properties - with a significant predominance of hydrophobicity. Its isoelectric point is pI 4.2. Fibroin is insoluble in water for this reason. Diluted solutions of many acids and alkalis, and becomes negatively charged at pH 7 [3]. Due to the structure of protein fibers, including natural silk fibroin, period of diffusion of the dye is about 1.0-2.0 hours at temperature up to 100 °C. The fibroin obtained from the Bombyx mori cocoons has a high specific surface area, and its fiber diameter is 15-20 microns.
The amorphous section of the fibroin is a structured medium with mobile part similar to a viscous liquid, but the possible forms of fluctuation cavities and slots are limited to an elastic frame [4].
Fisetin is a crystalline dark yellow powder, well soluble in methanol and ethanol. UV: \ , , A:
max(ethanol)
258, 267, 321, 370 nm; + AlCl3/HCl: 232, 277, 431 nm; IR(KBr): IR spectrum has absorption bands at 3000 and 2850 cm"1, corresponding to stretching vibrations of C-H link, at 1600, 1560 and 1510 cm"1, corresponding to stretching vibra-
tions of -C = C- aromatic system, bands at 1350 and 1260 cm1, corresponding to stretching vibrations of C-O, 3400 cm1; corresponding to stretching vibrations of phenolic -OH, 1050, 970, 900 cm1, bands of deformation vibration -C-H- substituted benzene ring and 1640 cm1, corresponding to stretching vibrations of C = O g-pyrone, 1425 cm1, corresponding to deformation vibration of C-H2, 3400-3300 cm1, corresponding to the stretching vibrations of the hydroxy groups [5]. Water solubility of fisetin is less than 1 mg/g.
Dyeing is one of the most complex and important process of natural silk product processing. In order to solve a certain part of this complex problem, we used the physical model of the distribution of the fisetin molecules in natural silk fiber. The key feature of this approach is that its microparameters - the process of the distribution of the molecules of fisetin in the medium of fibrin micro-fibrils, are subject of the macroparameters of the medium (density, temperature, concentration). To study kinetics of this process, we used the postulate of chemical kinetics - "The limiting stage principle". Seeing that in our case, diffusion process is divided into three successive stages connected in a certain way through the raw materials and intermediate compounds. The speed of entire process is determined by the diffusion rate constant, which is smallest (and limiting) in the third stage. Body of mathematics used in this work is based on a system of differential kinetic equations that determines the distribution functions of particles in a selected medium with selected speeds.
Experiment
To study the kinetic characteristics of fisetin molecules diffusion in microfibril fibrin, 4 g of fisetin was dissolved in 4 liters of distilled water and the reference solution was prepared in the same volume and at the same temperature from distilled water. Natural silk fibroin (cocoon thread) was thoroughly cleaned and dried to a constant weight of 100g and added into the process solution and into the refer-
ence solution. The dyeing process was carried out with constant stirring, so that the entire surface of the adsorbent was available for adsorption and excluded from consideration the uneven distribution of the substance in the volume. Thermostatic control was used to maintain a constant temperature. The experiment was carried out at a temperature of 293, 313, 333, 353, and 373K and at NaCl electrolyte concentration ofl, 2, 3, 4, 5 and 6g/l. Every 150 seconds, 20 ml samples were taken from the process solution and reference solution and distilled water was immediately added at an appropriate tempera-
ture to keep the solution volume constant. The concentration offisetin in the solution was measured by spectrophotometer at a wavelength of A = 313.3nm.
Results and discussion
Figure 1 shows the relation between the diffusion magnitude of fisetin in fibroin fiber and the duration of the treatment of fibroin with a dye at different temperatures, i.e. absorption isotherms: 1-293K; 2-313K; 3-333K; 4-353K and 5-373K. As is seen from this, the adsorption isotherms are at first form a straight line and can be characterized by their saturation.
i
S hb
ii
I J
£ '-ir
0.5
0.4
0.2
0.1
0
™>5 —
t fz ^04-
jpr / y v O < * j-
\3M jff O n / j 4 i t 2_
■ f ^— t —• *
a J >-1 —
J J*
_L
- ii
30
10 20 Treatment time, min
Figure 1. The relation between the diffusion magnitudes of fisetin in fibroin fiber and the duration of the treatment of fibroin with a dye at different temperatures: 1-293 K; 2-313 K; 3-333 K; 4-353 K and 5-373 K
The introduction of neutral electrolyte NaCl into the fisetin dye solution drastically reduces the potential barrier, which makes the dye anions approach the microfibrils to a mutual attraction distance, and the dye is adsorbed by fibroin fiber. NaCl solution was used as a neutral electrolyte.
The obtained result (Fig. 2) show that fibroin molecules are bound to Na+ and Cl- ions. Therefore neutral electrolytes not only reduce the potential barrier, but also compensate the electric charge, which results in better fiber adsorption. Figure 2 show that the concentration ofin the fisetin solution has an optimal value of - 5 g/l.
Experiments have shown that an excess of neutral electrolyte causes aggregation of fisetin anions, preventing their further diffusion into fibroin microfibrils. In summary, it was confirmed that the presence of neutral electrolytes helps to increase the adsorption of the dye fibers.
The adsorption process of fisetin by fibroin occurs almost instantaneously, and the diffusion of dye molecules into the inner fiber is interconnected with this process. In the process of actual dyeing, they cannot be separated. Therefore, the first and second stages of diffusion were studied by us as one, and the third stage was studied separately.
Number of specific dyeing issues were solved by using the physical model of the diffusion of the dye in the fibers and mathematical formulas (Fig. 3), that describe the kinetics of diffusion of the dye from the adsorption layer to the establishment of an equilibrium concentration in the entire volume of the fiber.
Mixing of substances that helps to balance the concentration occurs when there is a concentration gradient in solution. This is three-dimensional diffusion process. We use theoretical assumptions [6] to find a solution suggesting that if the concentration gradient exists only in one direction, then the diffusion issue can be perceived as a one-dimensional problem.
Figure 2. The relation between the content of fisetin molecules and the concentration of NaCI in fibroin at different temperatures: 1 -293 K\ 2-313 K\ 3-333 K\ 4-353 K\ 5-373 K
which the change in substance concentration occurs, and this amount (or mass) is proportional to the concentration gradient dC / dx, area aS and time At:
dC
Am = - D
v
dx
AS At,
(1)
Jt
( dC
v dx
- is
T
Figure 3. Three-dimensional physical model of the dye distribution in the fiber at the second stage of dyeing
As a part ofdiffusion process, an amount (or mass) ofthe substance An (or Am) in a definite time At, passes through area aS, located along the normal axis, along
where D - is the diffusion coefficient, -D
the flux density of a penetrating substance (this means the amount of a substance passing through a unit of area per unit of time). With D directly proportional to u - the average molecules velocity, and X - the average path of molecules:
(2)
D = UI.
Where dC / dx - is the concentration gradient of the solute (fisetin in our case) directed to x - center of the fiber.
It should be noted, that concentration C - is means a quantity that is numerically equal to the amount (or mass) of a given substance An (or Am) to the volume V of the mixture and C is expressed in any suitable units, such as mol/cm3 and g/cm3. In our case: SI - [n] = 1/m3{\displaystyle [n] = 1/mA{3}|; CGS {\displaystyle [n] = 1/cmA{3}}- [n] = 1/cm3.
Equation (1) formalizes Fick's first law and according to this Fick empirical equation, diffusion flux (J) of penetrant passing through sectional area is determined by the following equation:
'dCA
J = -D
dx
(3)
Jt
where{\displaystyle n={\frac {N}{V}}} J - is the diffusion flux is in the following units - mol/(cm3 x s) [7]. The negative sign in equation (4) appears due to the fact that the particles move in the direction of decreasing concentration.
By plugging (2) in the equation (3), we acquire the following (4):
j u - k r dC ^ (4)
T
The amount of dQx dye (fisetin), diffusing into the fiber (fibroin fiber) through the outer surface S, in a lengthwise direction of fiber for an infinitely small period of time dt, can be reduced to the following equation:
dQx = JSm0dt, (5)
where m0 - is the mass of fisetin molecules. From (3) and (5) we determine the following:
dQ = - D ( C
dx
Sm0dt.
(6)
Using equation (6), we studied the diffusion of fisetin in fibrin fibers. It was assumed that the distribution of the dye concentration over the depth of the fiber is linear, and this makes it possible to compose the following equation:
—=Ck, (7)
dx x
where Cm - is the equilibrium concentration of the dye until its fiber moves. This is the end ofthe first stage (the process of absorption) and beginning of second stage.
The equilibrium is concentration of the dye until its fiber moves. This is the end ofthe first stage (the process of absorption) and beginning of second stage.
To study the diffusion of the molecules of fisetin in the inner fibroin fiber a three-dimensional physical model of the dye distribution in the fiber at the second stage of dyeing was used (Fig. 3). It shows a cross section of fiber with unit length and the diffusion process reflected in the concentration scale. Therefore, fiber section with unit length has volume: V = nrl and side area: S = 2nr0.
Considering that the change in concentration during the transfer of dye through the side area over a period of time dt will be equal dCf and in the second stage the amount of penetrating dye in the direction x equals
dQx = ntfm0dCf. (7a)
Adding (7a) to (6) the following equation is ob-
tained:
nr^mdC'1 = - D
dC
nr^mdC'1 = - D
dx
Sm0dt, hah
dx 2nrmdt
(7b)
Equation (7) is written after the assumption that the dependence of the concentration of the dye from x is linear. Taking into account the fact that the direction of the fibroin fiber ro and the direction of diffusion x are opposite, instead of dC / dx gradient, we can add CM / x and compose (7b) as follows:
Figure 4. Model of changes in the volume concentration of dye directed to the center of the fiber
C
c
2 D, =---dt.
x ro
(8)
The resulting last expression (8) is the differential equation of the dyeing kinetics in the second stage of the process. To solve this equation, we first
integrate the expression (7a):
Qx =nr^m0 Cf + q,
where c1 - is the integral constant.
As the initial conditions, we assume that at the initial moment of dye contact with the fiber (t = 0), the fiber has no dye: Qx = 0 and, consequently, c1 = 0 . Therefore:
Qx =nr>o CI1. (9)
As shown in figure 4, we use the model of the second stage of fiber dyeing. This model represents the magnitude of the change in aV - the volume concentration of dye directed to the center of the fiber, depending on the size of x (dark areas). It is evident that:
x2
K--
3 y
By multiplying aV by m0 (the molecular weight of the dyes in this volume) and we find the amount Qx (mass) of the dye that has already penetrated into the inner part of the fiber:
AF = V,., -Vy =n-C
cyhndey ty. cone a
Qx =n-c m
.2 A
x 3
(9a)
According to the expression (9) and (9a) we can conclude that:
r2C" = C
'o ^ t ^x
2 \
x 3
or CMx2 - 3C j,x + 3Cfr2 = 0. (10)
Equation (10) is a quadratic equation in x and it has
since Cx > C?,
x = 1.5rn
+ im. Cî
v 3 c
V ' M y
two real solutions: . But x < r 0. Therefore, we
choose the following solution:
x = 1.5r„
(ii)
1 - 1 - 4 ■ Cl
V 3 C
V ' M J
By plugging (11) into equation (8), we get the
following equation:
i-,i-1 • ^
V 3 C
V ' M /
dC
C
4 D
=---^ dt.
3 r2
(12)
After integrating the differential equation (12), as integrals of irrational functions, the following equation is acquired:
\
i - 4 ■ —
3C
n \
c;1 8 D
<X> J
C
= 7 —t + C2 3 rn
(13)
To determine the integration constant c2, the initial time of the second stage of the diffusion process c2 is combined with the point of the corresponding end of the first stage tT. Therefore at t n = 0, the
C
CI
ratio Cr can be reduced to —-, which means:
^ = ^. (13 a)
CC
By putting equality (13a) into equation (13), we get the following equation:
\
4C
1 - 4. —1_
3 Cœ,
C
+ 2 —-, C „
where Cl - is the concentration of the dye inside the fiber at the end of the first stage of the diffusion process. If we plug the expression c2 into (13) we get the following equation:
\
( 4 Cf^
1 --
3 C
+ 2
» j
CL=8D
C " 3 r2
-t+
r 4 C1 v i -4 Ct
3C
» j
After simplification:
C
D
-t- = a; -t = p} C r2
4C
1 -4.
. 3 C»y
C
+ 2 —-C „
C
+ 2 —¿.(14)
C
= X ,(14 a)
and by plugging them into (14), we get the following equation:
1 - 4 •a
, 3 y
\3
-2a = 3 P + x .
(15)
In its canonical form (13), in relation to a, we get the following equation:
3 9 2 27 a--a '
16
16
i- 3V-x
V+9 27 x2 - 27
4 64 64 ^ And (16) is reduced to:
aa3 + ba2 + ca+ d = 0,
a +
= 0
(16)
(16 a)
3
where a = 1,
b = --9,
16
c = -
27 16
1 -3p-x
d=ip2+9 px+27 x - 27.
4 64 64 The cubic equation (16a) can be reduced to a canonical form by replacing the variable ^display-style x=y-{\tfrac {b}{3a}},} a = y - — that changes the equation form to:
(17)
bc
y + py+q = 0,
c b2 3ac - b2 2b3 where: p =---- =-— ; q =-----
a 3a2 3a2 27a3 3a2
2b3 - 9abc + 27a2d
d a
27a3
It is to be recalled that during the adsorption process, the equilibrium concentration in the adsorption layer is reached almost instantly: 11 = 0, C] = 0 (the concentration of the dye in the fiber at the end of the first stage ofthe diffusion process is zero). From (14 a)
we determine x = 1. Given this: a = 1; b =--;
9 9 16
c = —p and d = 3fi2 + -/3. Given that:
2 4
3c - b2
p = —— = -4.5/3 - 0.10547 and
q=-
2b3 -9bc + 27d
3
27
= 3fi2 + 1.40625,0- 0.0132. From this,
we calculate the discriminant of the cubic equation: = 2.25£4 -1.2657£3 + +0.317£2 - 0.0148^
f « > 3 f „ \
A = 1 p + i q i
I3 j 12 j
(18)
Considering R = — t, P> 0, (18) determines
r2
that A > 0 . If A > 0, then the cubic equation will have one real root and two conjugate complex roots [8]. We are only interested in the real root. The roots of the reduced cubic equation (17) can be found by the Cardano formula:
. „ A + B .A - B
y1 = A + B , y2,3 =— '
-± I-
-V3, (19)
Where A = ^-q + Va ; B = ^-q -JK and the
real root of the canonical equation (17) is y1 = A + B.
By plugging it into a = y-b, we find a for (16 a):
3a
^ = yi - b- = 3-q- . (20)
C M 1 3 v 2 v 2 3
The resulting mathematical relationship (20) is a kinetic equation that describes the first and second stages of the fiber dyeing process.
99
Considering that a = 1; b =--; c = — / h
2 9 16 2
d = 3B2 + — B is same for numerical calculation:
4
2
p = c - = -4.5 P - 0.1055 h
3
q = ■
2b3 - 9bc + 27d
27
= 3P2 +1.406P- 0.0264. (21)
Figure 5. Dependency graph of Cf/C concentration of dye molecules on time in r02/D unit of measurement
Figure 5 depicts graph of the kinetic dependence of Cf /C° on t - in a unit of r02 /D, constructed using equation (20) and formulas (21) (the minus sign is placed before the relative concentration, since the gradient of the dye concentration is negative). As graph shows, in the second stage of diffusion, the dependence of the relative concentration in time is parabolic, i.e. the process proceeds in accordance with the diffusion kinetics. This once again proofs the effectiveness of the chosen body of mathematics.
Conclusion. The kinetic equation of the first and second stages of the dyeing fibroin fibers dyeing process with fisetin using a physical model allows us to directly solve the problem of diffusion using numerical calculation. This approach has an advantage - the problem of diffusion does not require additional conditions when solving. This method allows you to fully describe the kinetic equation in both the first and second stages as one, as well as the third stage of the dyeing process, which is described in the next article.
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9. Гюнтер Н. М., Кузьмин Р. О. Сборник задач по высшей математике. Учебное пособие для вузов. -Санкт-Петербург: Лань, 2003.- 816 с.
