Diffusion coefficient calculated by complementary error function for the sublimation diffusion of disperse dye

Abstract

In order to find a simple and reliable method for the calculation of the diffusion coefficient, the correlation equation of concentration and distance in the form of complementary error function was derived from solving an ordinary differential equation of the diffusion equation. A disperse dye in paste was treated at 170°C–190°C for 3–4 h for the sublimation diffusion into polyethylene terephthalate using a film roll method. Quadratic regression analysis on the profile of dye concentration–distance was used to determine the surface concentration. The diffusion coefficient of each layer was calculated by obtaining the variable value of the complementary error function from the ratio of the mean dye concentration of each layer to the surface concentration. From linear regression analysis on the Arrhenius plot of the logarithm of the diffusion coefficient versus the reciprocal of absolute temperature, the correlation coefficient for the diffusion of 3 h was 0.9978 and that of 4 h was 0.9991. Thus, it was expected that the diffusion coefficients determined by the equation of complementary error function adopted in this experiment were reliable. The activation energy of diffusion for 3 h was 30.5 kcal mol⁻¹ and that for 4 h was 27.4 kcal mol⁻¹.

Keywords

Sublimation diffusion disperse dye polyethylene terephthalate error function regression analysis

Introduction

In the chemical wet process of the textile industry, high energy price and water inadequacy are the main cost factors. Improving energy efficiency and developing methods for water conservation must be the primary concerns for the dyeing and printing process of textiles. Commercial disperse dyes are usually classified according to their ease of transfer by sublimation for the processes of vapour-phase dyeing,¹ conventional printing^2,3 and sublimation transfer printing.^4,5 This is related to the their fastness to heat in hot processing. It is imperative that as much of the vaporised dye as possible should be absorbed by the polyester fibres.

The diffusion coefficient and the activation energy of disperse dyes are essential for the design of the sublimation processes. But, the methods to determine the diffusion coefficient reported in previous works are complicated and somewhat inaccurate. Sekido and Matsui⁶ applied a film roll method to study the kinetic behaviour of vat dyes in cellulose substrate. When the values of $\bar{C_{i}}$ (the mean concentration in ith layer from the outside) were determined colorimetrically, they obtained diffusion coefficient and surface concentration by calculating the values of $\bar{C_{i + 1}} / \bar{C_{i}}$ .^6,7 Sicardi et al.^8,9 applied a film roll method to measure the diffusion coefficients of dyes to polyethylene terephthalate (PET) film in supercritical carbon dioxide. They proposed a method of determining surface concentration by means of allowing a flap of film to protrude from the outer layer of the roll and reaching this flap to dye saturation at the polymer–bath interface.

The main objective of this study is to find a simple and reliable method for the calculation of the diffusion coefficient for the sublimation diffusion of disperse dye to PET using a film roll method. The equation of the complementary error function to determine the diffusion coefficient was derived from the diffusion equation by solving an ordinary differential equation (ODE). A quadratic regression analysis on the dye concentration–distance profile was used to determine the surface concentration. When a disperse dye in paste was treated at 170°C–190°C for 3–4 h, the diffusion coefficient was calculated with the ratio of the mean concentration of each layer to the surface concentration. Also, the activation energy of diffusion was calculated from the Arrhenius plot.

Material and method

Materials

C. I. Disperse Violet 26 (Sumicaron Bordeaux SE-BL) was used without purification. Biaxially oriented PET film (40 μm in thickness) was supplied by SKC. Sodium alginate (Duk San Pure Chemicals Inc.) was the extra pure grade and was used without purification.

Preparation of film roll

Figure 1 shows the schematic image of film roll method. The PET film of 40 μm in thickness was wrapped tightly around a glass rod of 1 cm in diameter. Both sides of film roll were wrapped by aluminium tape of which thickness is 180 μm. Sodium alginate paste was prepared with 300g of dye and 700g of 5% sodium alginate stock thickening. Sodium alginate paste containing disperse dye was coated on the film roll between both barriers of aluminium tape and dried at 100°C for 10 min in dryer. The top of roll was fixed firmly by a thin aluminium tape and cotton yarn with glass rod for dividing layers.^10,11

Figure 1.

Image of film roll coated by paste containing dye.

Sublimation

The film roll coated with paste containing disperse dye was treated in laboratory curing chamber at 170°C–190°C for 3–4 h for the sublimation of disperse dye. The heat treatment was performed by curing at the constant set temperature for the set time. When the heat treatment was finished, the paste coated was immediately stripped from the surface of film roll without any cooling operation. And then, the surface of film roll was washed with water until paste and unfixed dye were completely removed. The film roll washed was unrolled and cut into the section pieces representing the successive layers.

Determination of dye concentration

The section piece of each film layer, of which the dimension was $4 cm \times 3 cm$ , was dipped in test tube including chlorobenzene of 20 mL for the extraction of dye. The extraction of disperse dye was performed with boiling chlorobenzene at 132°C until the fixed dye in film was completely extracted. The optical density of $λ_{m a x}$ for the dye extracted was determined using spectrophotometer (Agilent Cary 8454), and the dye concentration ${(mg / cm}^{3})$ was calculated by the calibration curve prepared.¹²

Results and discussion

Differential equation of diffusion

The diffusion of dye is the process by which dye is transported from one part of a system to another as a result of random molecular motions. Consider a collection of dye particles performing a random walk in one dimension with length $x$ and time $t$ . If the diffusion coefficient $D$ is a constant, equation (1) is obtained by Fick’s second law, where C(x,t) is the dye concentration at position x at time t^13,14

\frac{\partial C (x, t)}{\partial t} = D \frac{\partial^{2} C (x, t)}{\partial x^{2}}

(1)

The general solutions of equation (1) can be obtained for a variety of initial and boundary conditions provided the diffusion coefficient is constant. Such a solution usually has one of two standard forms. Either it is comprised of a series of error functions or it is in the form of a trigonometrical series.¹⁵ In this study, a series of error functions was used to solve equation (1).

To solve the partial differential equation of equation (1), let any solution of the diffusion equation be $C (x, t)$ with a initial condition of equation (2) and the boundary conditions of equations (3) and (4)

C (x, 0) = f (x) = C_{0}, 0 < x < \infty

(2)

C (0, t) = C_{1}, \forall t > 0

(3)

C (\infty, t) = C_{2}, \forall t > 0

(4)

Let $C (x, t) = G (p)$ to seek solutions of equation (1), where $p = x / \sqrt{4 D t}$ . $G (p)$ is a function of variable $p$ and a special solution as the form of equation (5)¹⁶

C (x, t) = G (\frac{x}{\sqrt{4 D t}}) = G (p)

(5)

Equation (6) is obtained by differentiating $p$ with respect to $t$

\frac{\partial p}{\partial t} = \frac{\partial}{\partial t} (\frac{x}{\sqrt{4 D t}}) = - \frac{1}{2 t} \cdot \frac{x}{\sqrt{4 D t}} = - \frac{p}{2 t}

(6)

The chain rule yields equation (7) by differentiating equation (5) with respect to $t$ and combining with equation (6). The left side of equation (5) is transformed to an ODE of $G (p)$

\frac{\partial C (x, t)}{\partial t} = \frac{d G}{d p} \cdot \frac{\partial p}{\partial t} = - \frac{p}{2 t} G^{'} (p)

(7)

Similarly, equations (8) and (9) are obtained by differentiating $p$ with respect to $x$

\frac{\partial p}{\partial x} = \frac{\partial}{\partial x} (\frac{x}{\sqrt{4 D t}}) = \frac{1}{\sqrt{4 D t}}

(8)

\frac{\partial C (x, t)}{\partial x} = \frac{d G}{d p} \cdot \frac{\partial p}{\partial x} = \frac{1}{\sqrt{4 D t}} G^{'} (p)

(9)

Thus, the right side of equation (1) can be reduced to ODE of $G (p)$ , that is, equation (10)

\frac{\partial^{2} C (x, t)}{\partial x^{2}} = \frac{d}{d x} (\frac{1}{\sqrt{4 D t}} G^{'} (p)) = \frac{1}{4 D t} G^{″} (p)

(10)

By combining equations (7) and (10) with equation (1) of the diffusion equation, equation (11) is obtained for all $t > 0$

\frac{p}{2 t} G^{'} (p) + \frac{1}{4 t} G^{″} (p) = 0

(11)

By algebra, this is equivalent to the second-order variable-coefficient ODE equation (12)

G^{″} (p) + 2 p G^{'} (p) = 0

(12)

Equation (12) becomes equation (13) by applying the differentiation of logarithm

\frac{G^{″} (p)}{G^{'} (p)} = \frac{d}{d p} \ln [G^{'} (p)] = - 2 p

(13)

Integrating both sides of equation (13) with respect to $p$ yields equation (14), where $α$ is a constant of integration

\ln [G^{'} (p)] = \int (- 2 p) d p = - p^{2} + α

(14)

By exponentiating both sides of equation (14), equation (15) is obtained, where $e^{α} = k$

G^{'} (p) = e^{- p^{2} + α} = k e^{- p^{2}}

(15)

Equation (16) is obtained by integrating equation (15)

\int_{0}^{p} G^{'} (q) d q = G (p) - G (0) = \int_{0}^{p} k e^{- q^{2}} d q

(16)

Thus, the solution of equation (12) has the form equation (17), where $β = G (0)$ is another constant of integration

C (x, t) = G (p) = k \int_{0}^{p} e^{- q^{2}} d q + β

(17)

Boundary conditions

From the boundary condition equation (3), that is, $C (0, t) = C_{1}$ , taking the limit $x \to 0$ yields $\lim_{x \to 0} p = 0$ and $β = C_{1}$ as equation (18)

C (0, t) = k \lim_{p \to 0} \int_{0}^{p} e^{- q^{2}} d q + β = β = C_{1}

(18)

From the boundary condition equation (4), that is, $C (\infty, t) = C_{2}$ , taking the limit $x \to \infty$ yields $\lim_{x \to \infty} p = \infty$ and equation (19)

C (\infty, t) = k \lim_{p \to \infty} \int_{0}^{p} e^{- q^{2}} d q + C_{1} = C_{2}

(19)

Determining the value $k$ requires a Gaussian integral of equation (20)¹⁷

\int_{- \infty}^{\infty} e^{- x^{2}} d x = 2 \int_{0}^{\infty} e^{- x^{2}} d x = \sqrt{π}

(20)

Therefore, by combining a Gaussian integral of equation (20) with equation (19), equation (22) is obtained from equation (21)

C (\infty, t) = \frac{k \sqrt{π}}{2} + C_{1} = C_{2}

(21)

k = \frac{2}{\sqrt{π}} (C_{2} - C_{1})

(22)

Thus, the solution of equation (17) is given by equation (23)

C (x, t) = \frac{2 (C_{2} - C_{1})}{\sqrt{π}} \int_{0}^{\frac{x}{\sqrt{4 D t}}} e^{- p^{2}} d p + C_{1}

(23)

Gaussian kernel and initial condition

The function $C (x, t)$ is a solution of the diffusion equation. If $C (x, t)$ satisfies the diffusion equation of equation (1), then the partial derivatives of $C (x, t)$ , that is, equation (24), is also a solution of the diffusion equation

\frac{\partial C (x, t)}{\partial x} = \frac{(C_{2} - C_{1})}{\sqrt{π D t}} e^{- {(\frac{x}{\sqrt{4 D t}})}^{2}}

(24)

However, the Gaussian kernel $S (x, t)$ of equation (25) is the fundamental solution of the diffusion equation with a special initial condition

S (x, t) = \frac{1}{\sqrt{4 π D t}} e^{- {(\frac{x}{\sqrt{4 D t}})}^{2}}

(25)

By the property of translation invariance for the diffusion equation and its solutions, if $S (x, t)$ is a solution of equation (1), then the function $S (x - y, t)$ of equation (26) is also a solution for any fixed number $y$

S (x - y, t) = \frac{1}{\sqrt{4 π D t}} e^{- {(\frac{x - y}{\sqrt{4 D t}})}^{2}}

(26)

$S (x - y, t)$ has the form of a time-varying Gaussian function of equation (27), where $σ = \sqrt{D t}$ and $μ = y$

G (x) = \frac{1}{\sqrt{2 π} σ} e^{- {(\frac{x - μ}{2 σ})}^{2}} = \sqrt{2} S (x - y, t)

(27)

A solution of the diffusion equation can be also obtained by taking the convolution of the Gaussian kernel $S (x, t)$ with $f (x)$ of the initial condition. The convolution $(S \circ f) (x, t)$ is defined as equation (28), which only represents the solution for $t > 0$ ¹⁸

\begin{matrix} C (x, t) = (S \circ f) (x, t) \\ = \int_{- \infty}^{\infty} S (x - y, t) f (y) d y \\ = \frac{1}{\sqrt{4 π D t}} \int_{- \infty}^{\infty} e^{- {(\frac{x - y}{\sqrt{4 D t}})}^{2}} f (y) d y \end{matrix}

(28)

Thus, the total solution of equation (1) is the sum of equations (23) and (28). But from the initial condition and equation (23), taking the limit $t \to 0$ yields $\lim_{t \to 0} p = \infty$ and $C_{2} = f (x)$ through equation (29)

\begin{matrix} C (x, 0) = C_{0} = \frac{2 (C_{2} - C_{1})}{\sqrt{π}} \int_{0}^{\infty} e^{- p^{2}} d p + C_{1} \\ = \frac{2 (C_{2} - C_{1})}{\sqrt{π}} \cdot \frac{\sqrt{π}}{2} + C_{1} = C_{2} = f (x) \end{matrix}

(29)

If one face $x = 0$ of film is kept at a constant concentration $C_{1}$ , the other $x = \infty$ at $C_{2}$ and the film is initially at a uniform concentration $C_{0}$ , both $C_{0}$ and $C_{2}$ are zero in this experiment. Equation (28) is ignored by $f (y) = 0$ due to $C_{2} = f (x)$ in equation (29) and $C_{2} = 0$ . Thus, equation (23) becomes equation (30)

C (x, t) = \frac{- 2 C_{1}}{\sqrt{π}} \int_{0}^{\frac{x}{\sqrt{4 D t}}} e^{- q^{2}} d q + C_{1}

(30)

Complementary error function

The normalised Gaussian function is equation (31)¹⁹

G (x) = \frac{1}{\sqrt{2 π} σ} e^{- {(\frac{x}{\sqrt{2} σ})}^{2}}

(31)

Substituting $x / (\sqrt{2} σ) = t$ yields $d x = \sqrt{2} σ d t$ , and integrating equation (31) from $- t$ to $t$ yields the error function of $x / (\sqrt{2} σ)$ , that is, equation (32)

\int_{- t}^{t} G (x) d x = \frac{2}{\sqrt{π}} \int_{0}^{\frac{x}{\sqrt{2} σ}} e^{- t^{2}} d t = \erf (\frac{x}{\sqrt{2} σ})

(32)

The complementary error function $erfc (x)$ equals one minus the error function and can be obtained from changing the integration section as equation (33)

erfc (x) = 1 - \erf (x) = \frac{2}{\sqrt{π}} \int_{x}^{\infty} e^{- t^{2}} d t

(33)

Equation (30) can be represented by equation (34) and be converted to the error function as equation (35)

C (x, t) = C_{x} = C_{1} (1 - \frac{2}{\sqrt{π}} \int_{0}^{\frac{x}{\sqrt{4 D t}}} e^{- q^{2}} d q)

(34)

\frac{C (x, t)}{C_{1}} = \frac{C_{x}}{C_{1}} = 1 - \erf (\frac{x}{\sqrt{4 D t}})

(35)

Finally, $C_{x} / C_{1}$ is given by the complementary error function of $x / \sqrt{4 D t}$ as equation (36)

\frac{C_{x}}{C_{1}} = erfc (\frac{x}{\sqrt{4 D t}})

(36)

The diffusion coefficients for the sublimation diffusion of disperse dye were calculated by the values of $C_{x} / C_{1}$ and $x / \sqrt{4 D t}$ in equation (36). In this experiment using a film roll method, the mean dye concentration of each film layer corresponds to $C_{x}$ . But in order to compute the diffusion coefficients, the surface concentration, that is, $C_{1}$ , must be determined.

Surface concentration

When the disperse dye in paste was treated at 170°C–190°C for 3–4 h, the profile can be obtained from plotting the mean dye concentration of the ith layer $(\bar{C_{i}})$ versus distance $(x)$ . Quadratic regression analysis on the profile was used to determine the surface concentration $(C_{1})$ . A quadratic function of the dye concentration at position $x (C_{x})$ in terms of $x$ was given in equation (37), where $a_{1}$ and $a_{2}$ are coefficients

C_{x} = a_{1} x^{2} + a_{2} x + C_{1}

(37)

Figure 2 shows the mean dye concentration–distance $(\bar{C_{i}} vs x)$ profile and its quadratic regression curve for the diffusion at 190°C for 3 h. The value of $C_{1}$ extrapolated from the regression curve was 87.9 mg cm⁻³. The coefficient of determination $(r^{2})$ , that is, the variation in $C_{x}$ , is 0.994. If $r^{2} = 1$ , all of the data points fall perfectly on the regression line.

Figure 2.

Dye concentration–distance profile and its quadratic regression curve for the sublimation diffusion of disperse dye to PET film by treating at 190°C for 3 h $(r^{2} = 0.994)$ .

Figure 3 shows the mean dye concentration–distance $(\bar{C_{i}} vs x)$ profile and its quadratic regression curve for the diffusion at 180°C for 3 h. The value of $C_{1}$ extrapolated was 61.7 mg cm⁻³ $(r^{2} = 0.998)$ .

Figure 3.

Dye concentration–distance profile and its quadratic regression curve for the sublimation diffusion of disperse dye to PET film by treating at 180°C for 3 h $(r^{2} = 0.998)$ .

Figure 4 shows the mean dye concentration–distance $(\bar{C_{i}} vs x)$ profile and its quadratic regression curve for the diffusion at 170°C for 3 h. The value of $C_{1}$ extrapolated was 42.2 mg cm⁻³ $(r^{2} = 0.993)$ .

Figure 4.

Dye concentration–distance profile and its quadratic regression curve for the sublimation diffusion of disperse dye into PET film by treating at 170°C for 3 h $(r^{2} = 0.993)$ .

Quadratic regression analysis is the fit equation for a set of data shaped like a parabola.²⁰ From above results for the sublimation diffusion obtained from treating at 170°C–190°C for 3 h, it was found that the quadratic regression curves fitted well to the data of the dye concentration–distance profiles and so it was expected that the quadratic regression analysis would be a simple and reasonable method for the determination of the surface concentration.

Figure 5 shows the mean dye concentration–distance $(\bar{C_{i}} vs x)$ profiles obtained from treating at 170°C–190°C for 4 h and their quadratic regression curves. The value of $C_{1}$ for the treatment of 190°C was 88.5 mg cm⁻³ $(r^{2} = 1.000)$ , that for 180°C was 60.1 mg cm⁻³ $(r^{2} = 1.000)$ and that for 170°C was 41.6 mg cm⁻³ $(r^{2} = 0.999)$ . As the treatment time increased from 3 to 4 h, the mean dye concentrations of middle layers were increased, which resulted in $r^{2} ≒ 1.000$ .

Figure 5.

Dye concentration–distance profiles and their quadratic regression curve for the diffusion of disperse dye into PET film by treating at 170°C, 180°C and 190°C for 4 h.

Diffusion coefficient

The variable $X$ , that is, $x / \sqrt{4 D t}$ , of the complementary error function equation (36) is determined with the ratio of the dye concentration at distance $x (C_{x})$ to the surface concentration $(C_{1})$ . The profile of $C_{x} / C_{1}$ versus $x / \sqrt{4 D t}$ can be obtained from replacing $C_{x}$ with the mean dye concentration of the ith layer $(\bar{C_{i}})$ .

Figure 6 shows the plot of $\bar{C_{i}} / C_{1}$ versus $x / \sqrt{4 D t}$ for the sublimation diffusion at 190°C for 3 h, where $C_{1}$ is 87.9 mg cm⁻³. If the gap of the mean dye concentration between adjacent layers is large, the value of $X$ determined from the graph of the complementary error function shown in the insert also becomes large due to $D$ being small. The inter-layer diffusion coefficient $(D)$ is calculated from the value of $x / \sqrt{4 D t}$ . The values of $X$ determined with $\bar{C_{i}} / C_{1}$ and the diffusion coefficients calculated with $X$ for eight layers are presented in Table 1. The mean of the inter-layer diffusion coefficients $(\bar{D})$ was 4.6550 × 10⁻⁷ cm² min⁻¹.

Figure 6.

Plot of $\bar{C_{i}} / C_{1}$ versus $x / \sqrt{4 D t}$ for the diffusion at 190°C for 3 h (insert: graph of $Y = erfc (X)$ ).

Table 1.

The values of $X$ and the diffusion coefficients according to $\bar{C_{i}} / C_{1}$ for the diffusion at 190°C for 3 h.

$x (cm)$	$\bar{C_{i}} (mg {cm}^{- 3})$	$\frac{\bar{C_{i}}}{C_{1}} (= \frac{C_{x}}{C_{1}})$	$X (= \frac{x}{\sqrt{4 D t}})$	$D (= \frac{x^{2}}{4 t X^{2}}) ({cm}^{2} \min^{- 1})$
0.002	77.7	0.884	0.1032	5.2197 × 10⁻⁷
0.006	53.1	0.604	0.3667	3.7174 × 10⁻⁷
0.010	33.5	0.381	0.6195	3.6194 × 10⁻⁷
0.014	21.5	0.245	0.8221	4.0282 × 10⁻⁷
0.018	13.1	0.149	1.0204	4.3218 × 10⁻⁷
0.022	9.2	0.105	1.1463	5.1160 × 10⁻⁷
0.026	5.4	0.061	1.3248	5.3498 × 10⁻⁷
0.030	3.4	0.039	1.4596	5.8673 × 10⁻⁷
$\bar{D}$	–	–	–	4.6550 × 10⁻⁷

Figure 7 shows the plot of $\bar{C_{i}} / C_{1}$ versus $x / \sqrt{4 D t}$ for the diffusion at 170°C and 180°C for 3 h, where $C_{1}$ for 180°C is 61.7 mg cm⁻³ and $C_{1}$ for 170°C is 42.2 mg cm⁻³. The points of $\bar{C_{i}} / C_{1}$ versus $x / \sqrt{4 D t}$ for both of 170°C and 180°C must be on the same curve of $erfc (X)$ . The gaps between $\bar{C_{1}} / C_{1}$ and $\bar{C_{2}} / C_{1}$ for 170°C and 180°C were relatively large due to the small values of $\bar{C_{2}}$ , which was resulted from the comparatively slow rate of diffusion.

Figure 7.

Plot of $\bar{C_{i}} / C_{1}$ versus $x / \sqrt{4 D t}$ for the diffusion at 170°C and 180°C for 3 h.

In the case of 180°C, the values of $X$ and the inter-layer diffusion coefficients for five layers are presented in Table 2. The mean of the inter-layer diffusion coefficients was 2.0446 × 10⁻⁷ cm² min⁻¹.

Table 2.

The values of $X$ and the diffusion coefficients according to $\bar{C_{i}} / C_{1}$ for the sublimation of 180°C and 3 h.

$x (cm)$	$\bar{C_{i}} (mg {cm}^{- 3})$	$\frac{\bar{C_{i}}}{C_{1}} (= \frac{C_{x}}{C_{1}})$	$X (= \frac{x}{\sqrt{4 D t}})$	$D (= \frac{x^{2}}{4 t X^{2}}) ({cm}^{2} \min^{- 1})$
0.002	49.5	0.802	0.1773	1.7671 × 10⁻⁷
0.006	28.5	0.462	0.5201	1.8483 × 10⁻⁷
0.010	13.8	0.224	0.8598	1.8787 × 10⁻⁷
0.014	7.46	0.121	1.0964	2.2644 × 10⁻⁷
0.018	3.46	0.056	1.3513	2.4644 × 10⁻⁷
$\bar{D}$	–	–	–	2.0446 × 10⁻⁷

In the case of 170°C, the values of $X$ and the inter-layer diffusion coefficients for four layers are presented in Table 3. The mean of the inter-layer diffusion coefficients was 1.0488×10⁻⁷ cm² min⁻¹.

Table 3.

The values of $X$ and the diffusion coefficients according to $\bar{C_{i}} / C_{1}$ for the sublimation of 170°C and 3 h.

$x (cm)$	$\bar{C_{i}} (mg {cm}^{- 3})$	$\frac{\bar{C_{i}}}{C_{1}} (= \frac{C_{x}}{C_{1}})$	$X (= \frac{x}{\sqrt{4 D t}})$	$D (= \frac{x^{2}}{4 t X^{2}}) ({cm}^{2} \min^{- 1})$
0.002	30.8	0.730	0.2440	1.2220 × 10⁻⁷
0.006	11.6	0.275	0.7719	1.2186 × 10⁻⁷
0.010	4.70	0.111	1.1269	1.4477 × 10⁻⁷
0.014	1.80	0.043	1.4310	1.5631 × 10⁻⁷
$\bar{D}$	–	–	–	1.0488 × 10⁻⁷

Figure 8 shows the distribution of the inter-layer diffusion coefficients at each layer and the mean diffusion coefficients for the diffusion at 170°C, 180°C and 190°C for 3 h. The standard deviation $(σ)$ for 190°C was 0.840 × 10⁻⁷ cm² min⁻¹, $σ$ for 180°C was 0.303 × 10⁻⁷ cm² min⁻¹ and $σ$ for 170°C was 0.171 × 10⁻⁷ cm² min⁻¹. The large value of $σ$ for 190°C indicates that the inter-layer diffusion coefficients are a little spread out from the mean diffusion coefficient.

Figure 8.

The distribution of the inter-layer diffusion coefficients at each layer and the mean diffusion coefficients for the diffusion at 170°C, 180°C and 190°C for 3 h.

Figure 9 shows the plot of $\bar{C_{i}} / C_{1}$ versus $x / \sqrt{4 D t}$ for the diffusion at 170°C, 180°C and 190°C for 4 h. The points of $\bar{C_{i}} / C_{1}$ versus $x / \sqrt{4 D t}$ for 170°C, 180°C and 190°C, of course, were on the same curve of $erfc (X)$ . The gaps between neighbouring $\bar{C_{i}} / C_{1}$ were not very large, which indicated that the rate of diffusion was comparatively uniform across the whole layers diffused by dye. It was assumed that the above result was responsible for the increase in the mean dye concentrations of middle layers due to the increase in the heat treatment time from 3 to 4 h.

Figure 9.

Plot of $\bar{C_{i}} / C_{1}$ versus $x / \sqrt{4 D t}$ for the diffusion at 170°C, 180°C and 190°C for 4 h.

Figure 10 shows the distribution of the inter-layer diffusion coefficients at each layer and the mean diffusion coefficients for the diffusion at 170°C, 180°C and 190°C for 4 h. The standard deviation $(σ)$ for 190°C was 0.357 × 10⁻⁷ cm² min⁻¹, $σ$ for 180°C was 0.143 × 10⁻⁷ cm² min⁻¹ and $σ$ for 170°C was 0.155×10⁻⁷ cm² min⁻¹. It was found that the inter-layer diffusion coefficients were comparatively close to the mean diffusion coefficient.

Figure 10.

The distribution of the inter-layer diffusion coefficients at each layer and the mean diffusion coefficients for the diffusion at 170°C, 180°C and 190°C for 4 h.

Activation energy

Table 4 presents the common logarithm of the mean diffusion coefficient according to temperature. The activation energy of diffusion $(E_{a})$ can be calculated with the logarithm of the mean diffusion coefficients by equation (38) derived from the Arrhenius equation, where R = 1.978 cal K⁻¹ mol⁻¹

E_{a} = 2.303 R \frac{d (- \log \bar{D})}{d (T^{- 1})}

(38)

Table 4.

Logarithm of the mean diffusion coefficient.

T (K)	$\bar{D} (3 h), {cm}^{2} \min^{- 1}$	$- \log \bar{D} (3 h)$	$\bar{D} (4 h), {cm}^{2} \min^{- 1}$	$- \log \bar{D} (4 h)$
463	4.6550 × 10⁻⁷	6.3321	4.5800 × 10⁻⁷	6.3391
453	2.0446 × 10⁻⁷	6.6894	2.2491 × 10⁻⁷	6.6480
443	1.0488 × 10⁻⁷	6.9793	1.2001 × 10⁻⁷	6.9208

The correlation between the common logarithm of the diffusion coefficient $(\log D)$ and the reciprocal of absolute temperature $(T^{- 1})$ is represented as a linear function. Theoretically, all data points of $- \log D$ versus $T^{- 1}$ should lie on the straight line. In previous study,¹¹ the Arrhenius plots for the diffusion coefficients calculated by $ζ = ε / \sqrt{4 D t}$ , where $ζ$ was obtained from applying the ratio of the mean dye concentrations between adjacent layers, showed a poor linear relationship, from which the reliability of the diffusion coefficient was in doubt. Thus, the need for a reliable method for the calculation of the diffusion coefficient was required.

Figure 11 shows Arrhenius plots and their linear regression lines for the diffusion of 3 and 4 h. The correlation coefficient of the linear regression line is a measure of the linear correlation between two variables. As shown in the Arrhenius plots for the diffusion of 3 h and 4 h, all data points of $- \log D$ versus $T^{- 1}$ lay approximately on the linear regression line. The correlation coefficient for the diffusion of 3 h was 0.9978 and that of 4 h was 0.9991. Thus, it was expected that the diffusion coefficients determined by equation (36) were reliable.

Figure 11.

Arrhenius plots and their linear regression lines for the diffusion at 170°C, 180°C and 190°C for 3 and 4 h.

The activation energy of diffusion which indicates the sensitivity of the diffusion rate to temperature was calculated by equation (38) with the slope of linear regression line. The activation energy for 3 h was 30.5 kcal mol⁻¹ as shown in equation (39) and that for 4 h was 27.4 kcal mol⁻¹ as shown in equation (40)

\begin{matrix} E_{a} = 2.303 \times 1.987 \times 6.6695 \times 10^{3} \\ ≒ 30.5 \times 10^{3} (cal {mol}^{- 1}) \end{matrix}

(39)

\begin{matrix} E_{a} = 2.303 \times 1.987 \times 5.9954 \times 10^{3} \\ ≒ 27.4 \times 10^{3} (cal {mol}^{- 1}) \end{matrix}

(40)

Conclusion

The equation of complementary error function was derived from solving an ODE of the diffusion equation to find a simple and reliable method for the calculation of the diffusion coefficient. A disperse dye in paste was treated at 170°C, 180°C and 190°C for 3 and 4 h for the sublimation diffusion into PET using a film roll method. Quadratic regression analysis on the profile of concentration–distance was used to determine the surface concentration. The diffusion coefficient was calculated by the values of $C_{x} / C_{1}$ and $x / \sqrt{4 D t}$ . From linear regression analysis on the Arrhenius plot, the correlation coefficient for the diffusion of 3 h was 0.9978 and that of 4 h was 0.9991. Thus, it was expected that the diffusion coefficients calculated by the equation of complementary error function adopted in this experiment were reliable. Then, the activation energy for 3 h was 30.5 kcal mol⁻¹ and that for 4 h was 27.4 kcal mol⁻¹.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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