Mechanics of Composite Materials, Vol. 38, No. 2, 2002

DIFFUSION MOISTURE SORPTION BY AN ISOTROPIC POLYMER MATERIAL IN ATMOSPHERE WITH STATIONARY AND NONSTATIONARY HUMIDITY

O. A. Plushchik and A. N. Aniskevich

Keywords: diffusion moisture sorption, nonstationary boundary conditions, generalized criteria, polyester resin Solutions to the Fickâ€™s diffusion equation with stationary and nonstationary boundary and nonequilibrium initial conditions for describing the moisture sorption under actual operational conditions are presented. The cases of stepwise and harmonic changes in the environmental humidity are considered. Generalized criteria for comparing the moisture sorption under nonstationary and stationary conditions are developed. A possibility of accelerated computer simulation of natural periodic changes in the atmospheric humidity is shown. The solutions obtained are approbated on a polyester resin.

1. Introduction The properties of many structural polymer materials considerably depend on the amount of the absorbed moisture (see, for example, [1-3]). For many isotropic polymer materials, diffusion is the governing mechanism of moisture sorption. The complexity of investigating the effects caused by moisture consists in the fact that the moisture sorption is often a lengthy process, during which the moisture distribution in a material and, hence the properties of the material, vary with time and are nonuniform over its cross section. To estimate the moisture state of a material at an arbitrary time, we must investigate and describe the kinetics of the sorption process in stationary moisture conditions. During storage and operation, materials are subjected to variable moisture actions. These can be the periodic daily or seasonal changes in the atmospheric humidity, which, to a high degree of accuracy, can be regarded as harmonic or stepwise changes caused by specific features of operation of particular polymer articles or simply by carrying them from the storage to operational conditions. To take into account these actions, solutions to the corresponding nonstationary problems of moisture sorption must be found. The computer simulation of a moisture sorption process with variable boundary and initial conditions was discussed, for example, in [4, 5]. However, the diversity of possible operational conditions makes it necessary to examine each particular problem separately. In this case, the exact analytical solutions allow us to analyze the generalized regularities of the processes under study. The experimental modeling of natural periodic oscillations in the humidity and in some other nonstationary operational conditions is associated with certain technical difficulties. To predict the effect of moisture under such conditions, simInstitute of Polymer Mechanics, University of Latvia, Riga, LV-1006, Latvia, e-mail: andrej@pmi.lv . Translated from Mekhanika Kompozitnykh Materialov, Vol. 38, No. 2, pp. 223-244, March-April, 2002. Original article submitted November 27, 2001.

0191-5665/02/3802-0149$27.00 ÂŠ 2002 Plenum Publishing Corporation

149

plified short-term tests, whose parameters can be determined from analytical solutions with regard for the sorption characteristics of a particular material, must be carried out. The purpose of this study is to investigate and analyze the sorption processes in isotropic polymer materials under nonstationary external conditions. To this end, we will — obtain and analyze the analytical solutions to the problems of moisture sorption for the cases of stepwise and harmonic changes in the atmospheric humidity, — derive the generalized dependences characterizing the sorption kinetics in nonstationary conditions and compare the moisture sorption in stationary and nonstationary processes, — consider a possibility of accelerated modeling of natural oscillations in the atmospheric humidity in laboratory conditions, and — verify the obtained analytical solutions on a polyester (PE) resin in corresponding control tests. 2. Analytical Solutions to the Sorption Problems 2.1. Moisture sorption in an atmosphere with a stationary humidity. There is a number of models (the classical Fick’s model, the Langmuir two-phase model, the so-called models of non-Fickian moisture sorption, etc.), which make it possible to describe the moisture and water sorption in polymer binders and composites, where different features of the structure of a material or of the diffusing substance itself can be taken into account. The simplest and most widespread model for describing the moisture sorption process is the Fick’s diffusion model, which satisfactorily describes the kinetics of moisture and water sorption in many polymer materials within certain temperature ranges (the corresponding summary data are given in [2]). We will consider sorption problems for a specimen in the shape of a plane plate with a thickness a, i.e., a one-dimensional sorption process (however, the following calculations can be easily generalized to a three-dimensional case). The diffusion coefficient D is assumed to be constant. Let us analyze the solution to the well-known Fick’s diffusion equation ¶C ( x, t ) ¶ 2 C ( x, t ) =D ¶t ¶x 2

(1)

with the stationary and equilibrium boundary and initial conditions C ( x; 0) = C 0 ,

C ( 0, a; t ) = C ¥ ,

where C ( x, t )is the relative weight concentration of the diffusing moisture at a point x at a moment t and C 0 and C ¥ are the initial and equilibrium moisture concentrations in the material, respectively. The solution to this problem is (see, for example, [6, 7]) C ( x, t ) = C ¥ - 2

é æ pk ö 2 ù (C ¥ - C 0 ) ¥ [1- ( -1) k ] æ pk ö ê-ç ÷ Dt ú . sin x exp ç ÷ å k p è a ø êë è a ø úû k =1

(2)

The integration of Eq. (2) over the volume (thickness) of the specimen allows us to obtain an expression for the relative weight content of moisture, w: w( t ) = w ¥ - 2

( w¥ - w0 ) p2

¥

[1- ( -1) k ] 2

k =1

k2

å

é æ pk ö 2 ù exp ê-ç ÷ Dt ú , êë è a ø úû

(3)

where w 0 and w ¥ are the initial and equilibrium moisture contents in the specimen, respectively. The sorption characteristics of the material, D and w ¥ , can be obtained by approximating the experimental data by expression (3) and using them to describe the sorption processes with nonstationary boundary and/or nonequilibrium initial conditions. 2.2. Moisture sorption at stepwise changes in the atmospheric humidity. Let us consider the process of moisture sorption at stepwise changes in the atmospheric humidity. Depending on the initial moisture state of the material, the atmospheric 150

1.0

Cd /C¥ 1 2

0.5

3

x, cm 0.05

0

0.10

0.15

0.20

Fig. 1. Distribution of moisture concentration at various desorption times t = 0 (1), 20 (2), and 80 h (3), calculated according to Eq. (4).

humidity at the first stage, and its subsequent stepwise change, four different combinations, namely sorption–desorption, sorption–sorption, desorption–desorption, and desorption–sorption, are possible. However, from the viewpoint of the mathematical statement of the problem and its subsequent solution, all these variants are equivalent. Therefore, we will study only one of them, namely the sorption–desorption transition. If desorption begins at a moment t 0 when an equilibrium moisture content is reached, this process can be described by solution (2), (3). If the saturation is not achieved (due to large sizes of articles and a limited time of sorption) and the moisture concentration is distributed nonuniformly over the cross section, the moisture desorption begins with nonequilibrium values of moisture content. To describe such a process, we must find the solution to Eq. (1) at the initial C d ( x; t = 0) = C 0 t ( x, t 0 ) and boundary C d ( x = 0, x = a; t ³ t 0 ) = C 0 d conditions, where t = t - t 0 is the current time of desorption. The initial distribution of moisture concentration, C 0 t ( x, t 0 ) = C ( x, t 0 ), is found from Eq. (2) at t = t 0 . An example of such a distribution is curve 1 in Fig. 1,

which is calculated for a PE-resin specimen 2 mm thick at t 0 = 168 h and D = 1× 10-3 mm /h [8]. The solution to the problem 2

with nonhomogeneous initial conditions across the plate thickness can be obtained by the method of separation of variables [9]: C d ( x, t ) = C 0 d +

2 ¥ [1- ( -1) k ] å k {(C ¥ - C 0 d )exp[ -l2k Dt] p k =1

- (C ¥ - C 0 )exp[ -l2k D ( t + t 0 )]}sin ( l k x ).

(4)

As is seen, at t 0 ® ¥ (i.e., the desorption starts as soon as the saturation is reached), expression (4) takes the form (2) and the solution is similar to that for the case of sorption. Integrating Eq. (4) over the volume of the specimen, we come to an expression for the relative weight content of moisture during desorption, w d : wd ( t ) = w0 d +

2 p

2

¥

å

[1- ( -1) k ] 2

k =1

k

2

{( w ¥ - w 0 d )exp[ -l2k Dt]

- ( w ¥ - w 0 )exp[ -l2k D ( t + t 0 )]}.

(5)

For describing the sorption–desorption processes on one time scale, we have the following expressions: w( t ) = w ¥ - 2

( w¥ - w0 ) p

2

¥

å

k =1

[1- ( -1) k ] 2 k

2

exp [ -l2k Dt ] , t < t 0 ,

151

w/w¥

1.0

1

0.8

0.6

5

0.4

4 3

0.2

Fo

2 0.1

0

0.2

0.3

0.4

0.5

Fig. 2. Normalized relative moisture content as a function of the Fourier criterion under sorption (1) and desorption (2-5) starting at various moisture–sorption stages: Fo = 0.01(2), 0.05 (3), 0.1 (4), and 0.15 (5). Solid lines — calculation by Eq. (6) and dashed lines — by Eq. (3).

wd ( t ) = w0 d +

2 p2

¥

[1- ( -1) k ] 2

k =1

k2

å

{( w ¥ - w 0 d )exp[ -l2k D ( t - t 0 )]

- ( w ¥ - w 0 )exp[ -l2k Dt ]},

t ³ t0.

(6)

Expression (4) makes it possible to describe the propagation of the moisture front within the specimen upon desorption beginning with nonequilibrium values of moisture content. Figure 1 shows the curves of moisture concentration at different times calculated according to Eq. (4). On the initial stage of desorption, the moisture concentration is distributed over the specimen cross section highly nonuniformly, which can lead to considerable differences in the material properties over the cross section. Gradually, these curves level off and an equilibrium is achieved. Now, a question arises: to what extent the degree of the nonequilibrium initial state affects the subsequent desorption kinetics? To answer this question, we will examine Fig. 2, which shows the calculated sorption (1) and desorption (2-5) curves in relative coordinates. The desorption processes begin at different arbitrarily chosen instants of time. The solid lines show the desorption calculated from Eq. (6) with regard for the nonuniform initial distribution of moisture concentration; the dashed ones, which are calculated from Eq. (3), correspond to the assumption that the same amount of moisture is distributed uniformly over the cross section. In the figure, the dimensionless Fourier criterion Fo = Dt a 2 is plotted on the horizontal axis and the relative moisture content, normalized to its maximum value, w w ¥ , on the vertical axis. This makes it possible to present the results in a form independent of the characteristics of a particular material and of the conditions of sorption tests. As follows from the figure, the maximum difference (50% and more) between the desorption curves calculated by Eqs. (5) and (3) exists in the case where the desorption begins at early stages of sorption and the Fourier criterion at the “turning point” is smaller than ~0.05. When the “turning point” is shifted to later stages of the sorption process, the difference between the desorption curves becomes smaller, and, if the desorption process begins at Fo »0.15, it makes only several percents. Thus, using directly the data given in Fig. 2 or performing additional calculations by formulas (3) and (5) for intermediate cases, we can estimate the effect 152

of the nonuniform initial moisture distribution on the kinetics of subsequent desorption for any materials, specimen sizes, and sorption conditions. 2.3. Moisture sorption under periodic changes in the atmospheric humidity. Let us consider the kinetics of moisture sorption under daily, annual, or other variations in the humidity of the surrounding medium, whose change with time can be described in a general form by a harmonic function of the type [10] j( t ) = A sin ( wt + Y ) + B,

(7)

where A is the oscillation amplitude, B is the stationary component of humidity, wis the variation frequency, w = 2p T, T is the oscillation period, and Y is the initial phase shift. We will seek the solution to Fick's equation (1) for the case of one-dimensional diffusion at the initial C ( x, t = 0) = C 0 and boundary C ( x = 0, x = a; t ) = AC sin ( wt + Y ) + BC

(7a)

conditions. Here, AC and BC are the moisture concentrations corresponding to A and B. It is assumed that the harmonic changes in the atmospheric humidity, (7), lead to respective harmonic changes in the moisture concentration on the surface of the plate, (7a). Strictly speaking, this assumption is valid only in the case where the sorption isotherm of the material obeys the Henry law, i.e., the moisture concentration in the material increases linearly with atmospheric humidity. Otherwise, the sorption isotherm of the material on the range [B - A, B + A] must be approximated by a linear function. We seek the solution to this problem in the form [9] C ( x, t ) = V ( x, t ) + G ( x, t ),

(8)

where the first addend is the solution to the problem with nonhomogeneous and the second one with homogeneous boundary conditions. Based on the boundary conditions, the function V ( x, t ) has the form V ( x, t ) = AC sin ( wt + Y ) + BC .

(9)

The function G ( x, t ) can be found, by solving the nonhomogeneous diffusion equation (the so-called equation with a source) with homogeneous boundary conditions, as a sum of two solutions: G ( x, t ) = H ( x, t ) + F ( x, t ),

(10)

where H ( x, t )is the solution to the nonhomogeneous equation with homogeneous boundary and initial conditions, F ( x, t )is the solution to the homogeneous diffusion equation with homogeneous boundary conditions and the initial conditions G ( x; t = 0) = G0 = C 0 - [ A Ñ sin (Y ) + BC ] . Using a modified method of separation of variables, after some transformations, we arrive at the general solution H ( x, t ) of the nonhomogeneous equation,

¥

H ( x, t ) = å

k =1

é Sk é1 ùù g sin( wt + Y ) + k cos( wt + Y )ú ú ê 2 êw 2 w ûú ê gk ë ê1+ 2 ú ê w ú sin ( l k x ) , ê Sk é1 ú ù gk sin Y + cos Y ú exp ( -g k t )ú êê g 2k ë w w2 û ê ú ê 1+ 2 ú w ë û

(11)

2 [1- ( -1) k ] AC w . kp The solution of the homogeneous diffusion equation with homogeneous boundary conditions and stationary initial conditions is well known and is a special case of solution (2). Thus, the expression for F ( x, t ) is

where g k = Dl2k and S k = -

153

F ( x, t ) =

2G0 p

¥

[1- ( -1) k ] sin ( l k x )exp[ -l2k Dt ] . k k =1

å

(12)

Now, substituting Eqs. (11) and (12) into Eq. (10) and then Eqs. (10) and (9) into Eq. (8), we find the general solution to the problem with variable boundary conditions, C ( x, t ) = AC sin ( wt + Y ) + BC -

2 ¥ [1- ( -1) k ] å k p k =1

ì é ùü l2 D ï êsin ( wt + Y ) + k cos( wt + Y ) - ú ï ï w ê úï AC ï ê æ úï 2 ö ï ï ´ í æç l4k D 2 ö÷ ê- ç sin Y + l k D cos Y ÷ exp ( -l2 Dt )ú ý sin( l k x ). úï k ÷÷ ï ç 1+ w ÷ê ç w2 ÷ø êë çè úû ï ø ï çè ï ï 2 ïî + (C 0 - AC sin Y - BC )exp ( -l k Dt ) ïþ

(13)

The integration over the volume makes it possible to find an expression for the relative weight content of moisture: w( x, t ) = A w sin ( wt + Y ) + B w -

2 p2

¥

[1- ( -1) k ] 2

k =1

k2

å

ì é ùü l2 D ï êsin ( wt + Y ) + k cos( wt + Y ) - ú ï ï w ê úï Aw ï ê æ úï 2 ö ï ï ´ í æç l4k D 2 ö÷ ê-ç sin Y + l k D cos Y ÷ exp ( -l2 Dt )ú ý . úï k ÷÷ ï ç 1+ w ÷ê ç w2 ÷ø êë çè úû ï ø ï çè ï ï 2 ïþ ïî + ( w 0 - A w sin Y - B w )exp ( -l k Dt )

(14)

Here, A w and B w are the values of the moisture content corresponding to A and B. It is easily seen that, at w= 0, we have the case of moisture sorption under stationary conditions and formulas (13) and (14) are reduced to formulas (2) and (3), respectively. With unlimited increase in the frequency of external oscillations, w ® ¥, the diffusion front has no time to penetrate into the material and expressions (13) and (14) are also reduced to relations (2) and (3). Let us analyze expression (14) in more detail. The first component, ... A w sin ( wt + Y ) + B w ... , actually determines a cyclic change in the moisture content of the material for a very fast sorption process (at a very high diffusion coefficient or in a very thin plate). The combination of the first and second components, é ù l2 D ... A w sin ( wt + Y ) + B w -... êsin ( wt + Y ) + k cos( wt + Y )ú ... , w ú ê û ë determines the kinetics of a steady-state sorption process at cyclic changes in the boundary conditions. The presence of the function cos ( wt + Y ) in the second component corresponds to the phase shift between the changes in the atmospheric humidity and the moisture content in the material. The third component, æ ö l2 D ... - çç sin Y + k cos Y ÷÷ exp ( -l2k Dt )... , w ç ÷ è ø 154

w/w¥ 1.0

6

0.8

3

0.6

D 0.4

E

5 1

2

B

I

F G

H

C 0.2

A

4

t, h 0

500

1000

1500

2000

2500

Fig. 3. Sorption curves under periodic changes in the atmospheric humidity (1-3) with different periods: T = 1 (1), 14 (2), and 90 days (3) at A w = 0.3 and B w = 0.5%, calculated by Eq. (14), and in an atmosphere with stationary numidity (4-6) at w 0 = 0 and w ¥ = 0.2 (4), 0.5 (5), and 0.8% (6),calculated by Eq. (3)

determines the duration of the transition to steady-state oscillations. The value of the multiplicand ...exp ( -l2k Dt )... varies from 1 to 0 and characterizes the completeness of the transition period. And, finally, the fourth component, ... ( w 0 - A w sin Y - B w )exp ( -l2k Dt )... , determines the kinetics of the sorption process under stationary conditions. Using Eqs. (13) and (14), we can calculate the distribution of moisture concentration and the moisture content in a plane specimen at any instant of time under periodically varying boundary conditions (atmospheric humidity). Figure 3 shows the model calculations according to Eqs. (3) and (14) for different coefficients w ¥ and different oscillation periods: T = 24 (one day), 336 (a fortnight), and 2160 h (three months), and Fig. 4 depicts the same for different amplitudes: A w = 0.075, 0.3, and 0.85%, at B w = 0.5%. As seen from the figures, the change in the moisture content of specimens with time differs greatly from the moisture sorption under stationary boundary conditions. The concept of moisture saturation in the sense of a stationary moisture content cannot be applied here. However, we can speak of a conditional stabilization, since the alternation of the sorption and desorption processes leads to a steady-state periodic process. The moisture content, after some transition period, oscillates about a certain equilibrium value — a stationary component equal to B w — within the interval [B w - A w , B w + A w ], whose bounds are shown by curves 4 and 6 in Fig. 3. As expected, the amplitude of changes in the moisture content is the greater, the higher the amplitude and the period of oscillations in the atmospheric humidity (see Figs. 3 and 4). The solution obtained allows us to estimate the effect of various oscillations in the atmospheric humidity on the sorption kinetics. For example, let us consider constant-amplitude daily and seasonal oscillations in the atmospheric humidity (see Fig. 3). The moisture content at daily oscillations, according to the given time scale, changes insignificantly and practically coincides with the stationary component (curve 5). Obviously, in this case, it is reasonable to describe the moisture–sorption process by the solution to the problem with stationary boundary conditions (3), where the equilibrium water content w ¥ takes some averaged value B w . 155

w/w짜 1.0

0.8

3 0.6

1

2

4

0.4

0.2

t, h 0

500

1000

1500

2000

2500

Fig. 4. Sorption curves (1-3) under periodic changes in the atmospheric humidity with different amplitudes: A w = 0.075 (1), 0.3 (2), and 0.85% (3), B w = 0.5%, and T = 14 days, calculated by Eq. (14). Sorption curve (4) in a stationary atmosphere at w 0 = 0 and w 짜 = 0.5%, calculated by Eq. (3).

It is easy to estimate the error caused by the replacement of the nonstationary process with a stationary one and to determine the thresholds for the oscillation frequency and amplitude at which this replacement is possible. Under seasonal variations, the amplitude of changes in the humidity increases, and the difference between the moisture content calculated from Eq. (14) and the stationary component is considerable. In this case, for the sake of maximum precision, the process of moisture sorption must be described by the solution obtained for nonstationary boundary conditions (14). A similar pattern is observed in Fig. 4: at low oscillation amplitudes (A w = 0.075%), the moisture-sorption curve practically repeats the classical sorption curve obtained by solving the problem with stationary boundary conditions (3) at w 짜 = B w . Obviously, at greater amplitudes, the nonstationarity of the boundary conditions must be taken into account. Figure 5 depicts the moisture concentration curves calculated by Eq. (13) for the different instants of time shown by the corresponding points in Fig. 3. The moisture front moves in various directions, which is responsible for the highly nonuniform moisture distribution over the cross section of specimens, especially at the initial stage of the sorption process (see Fig. 5). At steady-state variations in the moisture content, these curves slightly level off and the discrepancy in the values of moisture concentration at any instant of time becomes considerable only on the boundaries of the specimen. It is interesting to note that, for the extreme (upper and lower) points of the steady-state section of the sorption diagram, the curves on their mirror images (in this case, the reflection plane is the stationary component of moisture concentration, B w ) repeat each other or completely coincide, which is explained by the symmetry of the external oscillations about B w and by the linear approximation of the sorption isotherm, as mentioned earlier (see Fig. 5b). Let us consider in more detail some features of the sorption process with harmonically varying boundary conditions. As already indicated, the oscillation amplitude of moisture content during a steady-state process depends on a number of parameters, such as the sorption characteristics of materials (diffusion coefficient), the geometry (plate thickness), and the characteristics of external oscillations (period and amplitude). We should also note that the oscillations in the moisture content of the material lag in phase behind the external oscillation in humidity. In the most general case, we will again use the generalized dimensionless quantities. For steady-state oscillations in the moisture content, at the initial phase shift Y = 0, expression (14) can be presented in the form 156

1.0

C /C¥

1.0

a

C /C¥

0.8

0.8

0.6

0.6

E,I H

0.4

F

D B

0.4

b

G C

0.2

A

0.2

x, cm 0

0.05

0.10

0.15

0.20

x, cm 0

0.05

0.10

0.15

0.20

Fig. 5. Distribution of moisture concentration in the specimen cross section at various times under periodic changes in the atmospheric humidity at AC = 0.3%, BC = 0.5%, and T = 14 days, calculated by Eq. (13): a) the initial stage of the sorption process (points A, B, C, and D in Fig. 3) and b) a steady-state periodic moisture-sorption process (points E, F, G, H, and I in Fig. 3).

w( x, t ) - B w Aw

é ê ê 2 = sin ( wt ) ê1ê p2 ê ê ë

é ê ê 2 - cos ( wt ) ê ê p2 ê ê ë

ù ú ú ¥ [1 - ( -1) k ] 2 1 ú å æ l4 D 2 ö ú k2 k =1 ç 1+ k ÷ú çç 2 ÷÷ ú w ø è û

ù ú 2 ú ¥ [1 - ( -1) k ] 2 l D 1 k ú . å 2 4 2 æ l D ö w ú k k =1 ç 1+ k ÷ ú çç 2 ÷÷ ú w è ø û

Then, taking into account that p sin a + q cos a = r sin ( a + Q ) and r 2 = p 2 + q 2 , the amplitude of these variations can be expressed as é ê ê 2 2 ê rw = 1ê p2 ê ê ë

ù ú ú ¥ [1 - ( -1) k ] 2 1 ú å æ l4 D 2 ö ú k2 k =1 ç 1+ k ÷ú çç 2 ÷÷ ú w ø è û

2

157

1.0

rw

1.0

rC 5 4 3

0.5

0.5

2 1

S 0

1

2

3

4

5

x/a 0

1

2

3

4

5

Fig. 6 Fig. 7 Fig. 6. Normalized oscillation amplitude of moisture content rw in a material as a function of the criterion S. Fig. 7. Normalized oscillation amplitude of moisture concentration rC in a material as a function of the relative distance from the specimen surface at S = 0.005 (1), 0.1 (2), 0.2 (3), 1 (4), and 3 (5).

é ê ê 2 +ê ê p2 ê ê ë

2

ù ú 2 ú ¥ [1 - ( -1) k ] 2 l D 1 k ú . å 2 4 2 w ú æ ö k l D k =1 ç 1+ k ÷ ú çç 2 ÷÷ ú w ø è û

By analogy with the Fourier criterion, we introduce a criterion S = DT a 2 for periodic processes. The physical meaning of the criterion S consists in the fact that, for the given specimen of a given material, this quantity is directly proportional to the oscillation period. At the same time, it is a dimensionless generalized quantity. Therefore, the normalized oscillation amplitude of the relative moisture content in the material, rw , as a function of the criterion S , which is shown in Fig. 6, is of a generalized character. As seen from the figure, at S = 0, oscillations are absent (their amplitude is zero). At S » 3-5, the oscillations are rather slow, the moisture content in the material has time enough to track the changes in the surrounding medium, and the normalized amplitude is close to unity. Within the interval 0 £ S £ 3-5, a peculiar “self-damping” of oscillations occurs because they have no time to pass through the middle of the specimen, and the amplitude is in the interval 0 < rw < 1. By analogy with the normalized oscillation amplitude of the relative moisture content, we introduce a normalized vibration amplitude of concentration é ù ê ú ê ¥ [1 - ( -1) k ] sin ( l x ) ú 2 k ú rC2 = ê1- å ê p k =1 k æ l4 D 2 ö ú ç 1+ k ÷ú ê çç 2 ÷÷ ú ê w ø è ë û

158

2

1.0

rC

1.0

rC

1

1 2

0.5

4

3

0.5

4

S 0

0.1

0.2

0.3

0.4

0.5

S 0

1

2

3

4

5

Fig. 8. Normalized oscillation amplitude of moisture concentration rC in a material as a function of criterion S at x a = 0.1 (1), 0.15 (2), 0.25 (3), and 0.5 (4). The initial part of the curve is shown in the inset. 2

é ù ê ú 2 ú ê ¥ k [1- ( -1) ] sin( l k x ) l k D ú 2 . + ê å ê p k =1 k æ l4 D 2 ö w ú ç ÷ k ê ú çç 1+ 2 ÷÷ ê ú w è ø ë û Obviously, this amplitude depends on the relative distance x a from the surface of the specimen (the coordinate is normalized to the thickness of the specimen and varies from 0 at the edge to 0.5 at the middle of the specimen). We should note that, if the penetration depth of oscillations is defined as the distance from the surface at which the oscillation amplitude decreases by a fixed factor (10, 2. 718, etc.), this quantity does not depend on the amplitude of external oscillations. Figure 7 shows the normalized vibration amplitude of moisture concentration in the material, rC , as a function of the relative depth at different values of the criterion S. As seen from the figure, at large values of S (S »3), the vibration amplitude is approximately the same across the whole depth of the specimen, whereas, at small S (S » 0.005), it is close to zero on 3/5 of the thickness of the specimen. The normalized oscillation amplitude of concentration, rC , at a certain depth, depends on the diffusion coefficient of the material, the plate thickness, and the oscillation period. In a generalized form, such a dependence on the criterion S is shown in Fig. 8. The figure illustrates the initial “delay” of the vibration amplitude at the middle of a specimen and its rapid growth near the surface at small oscillation periods. It is also seen that, at large periods, the amplitude, in all sections of the specimen, tends to its maximum value equal to unity. 3. Experimental Approbation of the Solutions Obtained The analytical solutions were approbated on specimens made of a Norpol 440-Ì888 PE resin, which is based on an unsaturated orthophtalic PE resin and styrene in the amount of about 43 ± 2 wt.%. The specimens were cured by 1.5% methyl ethyl ketone peroxide (MEKP) at 25 ± 2°C. The moisture sorption and desorption were carried out in the atmosphere above a silica gel and saturated NaCl and K2SO4 solutions (at constant relative humidities of j = 0, 77, and 98%, respectively) and in water (conditionally, at j = 100%) at room temperature. The specimens were thin plane-parallel strips 2 ´ 10 ´ 150 mm in size. Figure 9 shows the experimental data for the PE resin on moisture sorption in an atmosphere with a stationary humidity [8] and the corresponding calculation results obtained by Eq. (3). The equilibrium moisture content as a function of the relative humidity (the sorption isotherm) is shown in Fig. 10. It is found that the diffusion coefficient does not depend on the moisture concentration and is D = 1× 10-3 mm2/h. As seen from the data in Fig. 9, the moisture sorption by the PE resin at room 159

1.5

w, %

1.6

3

1.2

2

1.0

w¥ , %

1

0.8

0.5 0.4

Öt, h 0

10

20

30

40

50

j, %

60

0

20

40

60

80

100

Fig. 9 Fig. 10 Fig. 9. Sorption curves for a PE resin in atmosphere at j = 77 (1, u), 98% (2, s), and in water (3, n ) at room temperature; dots — experimental data and lines — calculation. Fig. 10. Sorption isotherm for a PE resin; dots — experimental data and lines — calculation.

100

a

j, %

60

t, h

20 85

0 0.6

170

255

w, %

340

400

b 3

0.4

1

2

0.2

t, h 0

80

160

240

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Fig. 11. Changes in the atmospheric humidity (à) and sorption–desorption curves under a stepwise change in the humidity from j = 98 to 47% (b); dots — experimental data and lines — calculation by Eq. (6) (curves 1 and 2) and Eq. (3) (curve 3). temperature can be described by the classical diffusion model and Eq. (3) describes the experimental data quite satisfactorily. The discrepancy between the calculated and experimental sorption curves does not exceed 5% on the average. To approbate the analytical solution of the sorption problem under nonequilibrium initial conditions, a stepwise change in the atmospheric humidity was simulated experimentally. The control sorption—desorption experiment was performed according to the scheme shown in Fig. 11a. First, the specimens were wetted in a desiccator with an atmospheric humidity of j = 98%. At the instant of time t 0 = 168 h (within 7 days), when the moisture content reached w = 0.56% (for the distribution of moisture over the cross section of specimens, C ( x, t 0 ), see curve 1 in Fig. 1), the specimens were placed in a desiccator with j = 47% for 7 days. As seen from the data in Fig. 11b, the results of the control experiment are within the limits of the clas-

160

j, %

1

a

2

t, h 0

1.6

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1000

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2500

b 2

1.4 1.2 1.0

3

0.8

4

0.6

5

0.4 0.2

0

1

t, h 500

1000

1500

2000

2500

Fig. 12. Cyclic changes in the atmospheric humidity in the control experiment (1) and its approximation (2) by a harmonic function (a) and moisture sorption under periodic changes in the atmospheric humidity with T = 14 days (b); dots — experimental data and lines — calculation: by Eq. (14) at A w = 0.38% and B w = 0.65% (1); by Eq (3) at w 0 = 0 and w ¥ = 1.4 (2), 0.8 (3), 0.65 (4), and 0.4 % (5). sical diffusion model of moisture sorption and Eq. (6) describes the experimental data satisfactorily. Notice the visible discrepancy between the calculated desorption curves 2 and 3 in Fig. 11b. Curve 2 was obtained from Eq. (6), i.e., the desorption from nonequilibrium values of the moisture content was considered, and curve 3 was obtained from Eq. (3), i.e., the desorption after saturation was considered. The realization of an accelerated sorption experiment in laboratory conditions with harmonic changes in the atmospheric humidity, which models the natural atmospheric oscillations, is associated with certain technical difficulties. Therefore, an attempt was undertaken to replace the natural cycle of oscillations in the atmospheric humidity by a stepwise rectangular cycle with the same area. Such an experiment is easier technically and, in addition, allows one to reproduce a rather wide frequency band of oscillations (see Fig. 11a). However, the final goal of these tests was the computer simulation of harmonic oscillations, and the experimental results were then approximated by a function of type (7). This allowed us to greatly simplify both the analytical calculations and the realization of the experiment, and also to predict the sorption kinetics of any material with a sufficient accuracy. Specimens with an initial moisture content of w = 0 wt.% were placed in a desiccator at an atmospheric humidity of j = 98% for seven days (168 h) and then placed in a desiccator with j = 47% for seven more days. Such a stepwise rectangular cycle with a 14-day period, the scheme of which is given in Fig. 12a, was repeated eight times. The parameters of function (7), which approximates the experimental cycle, were determined from the data given in Fig. 10. In this case, as already mentioned, the section 161

B - A £ j £ B + A of the sorption isotherm was approximated by a straight line. In such a way, we found that A w = 0.38% and B w = 0.65% in Eq. (7) for the oscillation frequency of humidity w = 2p T (at T = 336 h) and Y = 0. Figure 12b presents the experimental data and the calculation results obtained by Eq. (14) for the values of the coefficients given above. As expected, the extreme points (the values of moisture content) corresponding to the jumps in the atmospheric humidity shown in Fig. 12a do not fall on the calculated curve, since the approximation of a rectangular cycle of changes in humidity by sine curve (7) introduce certain errors into our calculations. Nevertheless, the data in Fig. 12b show that the moisture-sorption process is periodic and a conditional saturation can be achieved at periodic changes in the atmospheric humidity; they also allow one to determine the stationary component of the moisture content for the PE-resin specimens of the given shape under known nonstationary moisture conditions. Thus, the replacement, in experiments, of a sine cycle by a rectangular one does not distort the information on the character of the moisture-diffusion process proceeding under nonstationary boundary conditions and, in addition, allows one to determine, with a satisfactory accuracy, the stationary component of the moisture content. Very often, precisely these data are most necessary for predicting the process of moisture sorption in materials operating under natural climatic conditions. Conclusion 1. Analytical solutions to the sorption problems with nonstationary boundary and nonequilibrium initial conditions, which describe the propagation of the moisture concentration front in actual operational conditions, are obtained and analyzed. 2. Generalized dependences characterizing the sorption kinetics under nonstationary conditions are obtained, and criteria for comparing the moisture sorption in nonstationary and stationary processes are elaborated. 3. It is shown that the natural periodic changes in the atmospheric humidity occurring according to a harmonic function can be modeled by a stepwise rectangular cycle, which makes it possible to significantly simplify the realization of experiments in laboratory conditions and to predict, with a sufficient accuracy, the character of sorption curves and the moisture content. 4. The solutions obtained were approbated on PE-resin specimens. The experimental data coincide with calculation results satisfactorily. The sorption characteristics of the material, found experimentally under stationary external conditions, allowed us to describe the moisture sorption in an atmosphere with nonstationary humidity. Acknowledgment. The study was financially supported by the Latvian Council of Science.

REFERENCES 1. G. S. Springer, Environmental Effects on Composite Materials. Vol. 3, Technomic Publishing Co. (1988). 2. Y. Weitsman, “Effects of fluids on polymeric composites. A review,” in: Comprehensive Composite Materials. Vol. 2, Elsevier Science Ltd. (2000), pp. 1-33. 3. O. A. Plushchik and A. N. Aniskevich, “Effects of temperature and moisture on the mechanical properties of polyester resin in tension,” Mech. Compos. Mater., 36, No. 3, 233-240 (2000). 4. W. P. De Wilde and P. Frolkovic, “The modelling of moisture absorption in epoxies: effects at the boundaries,” Composites, 25, No. 2, 119-127 (1994). 5. L. W. Cai and Y. Weitsman, “Non-Fickian moisture diffusion in polymer composites,” J. Compos. Mater., 28, No. 2, 130-154 (1994). 6. J. Crank, The Mathematics of Diffusion, Oxford (1956). 7. R. D. Stepanov and O. F. Shlenskii, Strength Calculations of Plastic Structures Operating in Fluid Media [in Russian], Moscow (1981). 8. O. A. Plushchik and A. N. Aniskevich, “Water sorption and swelling of polyester resin,” Mater., Tekhnol., Instrum., 6, No. 1, 49-53 (2001). 9. N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov, Basic Differential Equations of Mathematical Physics [in Russian], Moscow (1962) 10. Yu. S. Urzhumtsev and R. D. Maksimov, Prediction of Deformability of Polymer Materials [in Russian], Zinatne, Riga (1975). 162