11institutetext: Institut de Physique Théorique, CEA, IPhT, CNRS, URA 2306,
F-91191 Gif-sur-Yvette, France, 22institutetext: Faculty of Engineering,
Takushoku University, Hachioji, Tokyo 193-0985, Japan, 22email:
<EMAIL_ADDRESS>33institutetext: Department of Physics, Faculty
of Science, Kyushu University, 4-2-1 Ropponmatsu, Fukuoka 810-8560, Japan

# Kinetics of water flow through polymer gel

Yasuo Y. Suzuki 1122 Masayuki Tokita 33 Sada-atsu Mukai 33

(Received: date / Revised version: date)

###### Abstract

The water flow through the poly(acrylamide) gel under a constant water
pressure is measured by newly designed apparatus. The time evolution of the
water flow in the gel, is calculated based on the collective diffusion model
of the polymer network coupled with the friction between the polymer network
and the water. The friction coefficient are determined from the equilibrium
velocity of water flow. The Young modulus and the Poisson’s ratio of the rod
shape gels are measured by the uni-axial elongation experiments, which
determine the longitudinal modulus independently from the water flow
experiments. With the values of the longitudinal modulus and of the friction
determined by the experiments, the calculated results are compared with the
time evolution of the flow experiments. We find that the time evolution of the
water flow is well described by a single characteristic relaxation time
predicted by the collective diffusion model coupled with the water friction.

###### pacs:

83.10.BbKinetics of deformation and flow and 83.80.KnPhysical gels and
microgels

##  1 Introduction

The gel is an important state of matter that is found in a wide variety of
biological, chemical, and food systems TanakaSci . The polymer gel consists of
a cross-linked polymer network and a large amount of solvent (typically
water). Since the average mesh size (the distance between neighboring
crosslinks) of the polymer network is in general large compared to small
molecules, they can pass through the gel easily. Two different transport
processes are important in the gel.

The first one is the diffusive flow of molecules Muhr . The transport of
molecules by the diffusion in the polymer network is modeled as the diffusion
of the probe objects in the fixed mesh of obstacles. The ratio between the
probe size and the mesh size plays essential role to determine the diffusion
coefficient of the molecules in the gels. The diffusion of the probe molecules
in the gel has been measured by the pulsed field gradient nuclear magnetic
resonance. The results indicate that the diffusion coefficient of the probe
molecules of various sizes in the gel is well described by a simple scaling
relationship TokitaPRE .

The second process is the convective flow of the water through the polymer
network TokitaJCP95 ; TokitaSci ; TokitaAPS . When the water flows in the gel,
it experiences the hydrodynamic friction from the polymer network, at the same
time, the polymer network of gel is deformed from the initial configuration by
the drag force of the water. The entire flowing process of the water is
determined by the balance between the viscoelastic response of the water flow
and that of the polymer network. The collective diffusion of the polymer
network coupled with water friction, therefore, plays essential roles in the
convective flow process in the gel.

As well as the transport phenomena in the gel, the water flow is also
important in the kinetics of the volume phase transition of the gel including
the pattern formations in shrinking gelsTanakaPRL ; TanakaJCP70 ; Matsuo ;
TokitaJPSJ ; MaskawaJCP ; TakigawaJCP111 ; TokitaJCP113 ; TakigawaJCP117 ;
BoudaoudPRE ; UrayamaJCP ; NosakaPoly . Kinetics of solvent flow in the gel is
the main subject of the application of the gel to control the solvent flow
YoshikawaJJAP ; SuzukiJCP . Although the significance, the solvent flow
process in the polymer gel has yet to be studied in detail.

It is the purpose of this paper to describe the kinetics of the water flow
through the polymer gel, based on the collective diffusion model coupled with
water friction. The characteristic relaxation time of the water flow process
in the gel is expressed as $\tau=L^{2}/\pi^{2}D_{c}$ with the parameters: the
typical size of the gel, $L$, and the collective diffusion coefficient of the
polymer network, $D_{c}$. The collective diffusion coefficient is expressed by
using the longitudinal modulus of the gel, $\kappa$, and the friction
coefficient between the polymer network and the water, $f$, as TanakaJCP70 ;
TanakaJCP59

$D_{c}=\frac{\kappa}{f}.$ (1)

The water flow in the polymer network is, thus, directly governed by the
viscoelastic property of the gel. The collective diffusion coefficient of
polymer gels has been extensively measured by quasi-elastic light scattering
Munch1 ; Munch2 . The friction coefficient between the polymer network of gel
and the water has been measured by flowing the water through the gel
TokitaJCP95 ; TokitaSci ; TokitaAPS .

In this paper, we present experimental results on the water flow through the
gel, and show that the time evolution of the water flow through the gel is
described satisfactorily by the collective diffusion model of polymer network
with water friction. The information on the kinetics of water flow in the gel
will help the better use of them in the applications.

##  2 Experiment

###  2.1 Water flow measurements

The experimental setting of the water flow measurements is illustrated
schematically in Figure 1.

Figure 1: Experimental setting of water flow measurements

A thin circular slab gel of thickness, $L$, is set in a tube and the rim of
the gel is glued to the tube. The surface of the gel at the lower reaches side
is mechanically fixed to the end plate through an O-ring. When the small
pressure, $P_{0}$, is applied to the water from the upper stream side of the
gel, the water flows through the gel deforming the polymer network of gel by
the drag force. The macroscopic value of the friction coefficient of the gel,
$f$, is determined from the velocity of the water flows out of the gel in the
equilibrium state, $v_{\infty}$, as

$f=\frac{P_{0}}{v_{\infty}L}.$ (2)

Even though the concept is simple, the mechanical measurement of the friction
coefficient between the polymer network of gel and the water is not easy
because the friction of the gel is huge. The difficulty of evaluating the
friction coefficient measured from the water flow through the gel is described
in the reference TokitaJCP95 . In our experiments, the difficulty is overcome
by using a well calibrated homogeneous glass capillary to amplify the velocity
of water as shown in Figure 1. The position of the meniscus of water in the
capillary is measured as a function of time after the pressure is applied to
the water.

The photographs of the apparatus used in this study are given in Figure 2.

Figure 2: Apparatus for water flow measurements: (a) the upper photograph
shows the external appearance, (b) the lower photographs shows the inside of
the end plates.

The essential structure of the apparatus is the same with the previous one but
some points are improved for the precise measurements of the friction of the
gel TokitaJCP95 . The right hand side of Figure 2 (a) is the upper stream side
and is connected to a water column to apply the hydrostatic pressure. The
glass capillary is set at the top of stainless steel pipe which can be seen at
the left hand side of Figure 2 (a). Leak of water around the gel is prevented
by setting O-rings inside the cell as shown in Figure 2 (b). All parts of the
cell is made of stainless steel to avoid the mechanical deformation due to the
applied pressure. The mechanical deformation of the apparatus is considerably
reduced compared to the previous one.

###  2.2 Mechanical response measurements

The mechanical response of the polymer gel relates the friction coefficient to
the collective diffusion coefficient of the polymer network as
$D_{c}=\kappa/f$ where $\kappa=K+\frac{4\mu}{3}$ TanakaJCP59 . Here, $\kappa$
is the longitudinal modulus of the gel and $K$ and $\mu$ are the bulk modulus
and the shear modulus of the gel, respectively. We measure two elastic moduli
of the gel simultaneously, and the longitudinal modulus of the gel is
determined from them.

The rod shape gel is elongated in water at a strain of about 10 $\%$ and the
stress is measured. The size of the gel is observed by using a microscope
during the elasticity measurements to obtain the Young’s modulus of the gel,
$E$, and the Poisson’s ratio, $\sigma$. The longitudinal modulus of the gel,
$\kappa$, is determined from the Young’s modulus and the Poisson’s ratio by
$\kappa=K+\frac{4\mu}{3}$, $K=E/3(1-2\sigma)$ and $\mu=E/2(1+\sigma)$.

In this short time experiments, the water solvent neither goes out nor comes
into the gels. Thus we determine the mechanical response of the gel
independently from the friction experiments by the water flow. We use the
results to analyze the kinetics of water flow through the polymer network.

###  2.3 Sample

The samples used in this paper are poly(acrylamide) gel that is obtained by
the radical co-polymerization of the main-chain component, acrylamide, and the
cross-linker, N,N’-methylene-bis-acrylamide. The concentration of the gel used
in this measurement is 1 M. Ammoniumpersulfate and N,N,N’,N’-tetramethyl-
ethylenediamine are used as the initiator and the accelerator. All chemicals
used here are of electrophoresis grade, purchased from BioRad and used without
further purification. The gel for the friction measurements is prepared in the
circular gel mold, where the gel-bond films (FMC) with circular opening are
glued on the both sides. For the friction coefficient measurements, the gel is
prepared in the measurement cell and kept under water at a constant room
temperature for overnight to reach the equilibrium state.

For the elastic modulus measurements, the rod shaped gels are prepared in the
capillary. The desired amounts of the main-chain component, the cross-linker,
and the accelerator are dissolved into the distilled and de-ionised water,
which is prepared by a Mili-Q system. The pre-gel solution is de-gassed for 20
min, and then the desired amount of the initiator is added to the solution to
initiate the reaction. The sample gels are taken out of the reaction bath and
then washed by distilled and de-ionized water extensively and used in each
measurement.

##  3 Experimental results

###  3.1 Friction coefficient of gel

The experimental results of the measurement of water flow through the gel are
shown in Figure 3.

Figure 3: Position of meniscus in capillary fit by equation (19) with $a$ and
$b$ as adjusted parameters.

The position of the meniscus in the capillary of $10\mu l$ glass micro-pipette
is plotted as a function of the time elapsed after the application of the
pressure to the water. The slope of the flow curve, thus obtained, represents
the water velocity in the capillary, $v^{\prime}$. The velocity of the flow is
high at first. Then it eventually decreases with time and approaches to an
equilibrium value. The equilibrium state is attained about 5 to 7 $\times$ 103
s after the pressure is applied to the water. The velocity of the meniscus in
the equilibrium state is determined by the least-squares analysis of the
linear portion of the flow curve in Figure 3, typically in the time region
more than 1 $\times$ 104 s, that yields to $v^{\prime}_{\infty}=1.9\times
10^{-7}$ m/s. The velocity of water flow in the gel, $v$, is then calculated
from the velocity of the meniscus, $v^{\prime}$, with the ratio of the radius
of the circular opening of the gel mold, $R$, and that of the capillary, $r$,
as $v=v^{\prime}(r/R)^{2}$. The values of $R$ and $r$ are measured as 1.745 mm
and 0.268 mm, respectively. The equilibirium velocity of water flow in the gel
is, thus, calculated as $v_{\infty}=4.4\times 10^{-9}$ m/s. The pressure
applied to the water is $P_{0}=2.9\times 10^{3}$ N/m2, which corresponds to
the height of water column of 30 cm, and the thickness of the gel is
$L=1.0\times 10^{-3}$ m. From these values, the friction coefficient of the
gel is calculated by equation (2) that yields to $f=6.6\times 10^{14}$ N s/m4.
This agrees with a typical value of the friction coefficient of the
transparent poly(acrylamide) gel evaluated by other experiments TokitaJCP95 .

###  3.2 Elastic modulus of gel

In Figure 4, we show the concentration dependence of the Young’s modulus and
the Poisson’s ratio of the gels used in our experiment.

Figure 4: Concentration dependence of elastic moduli: (a) the upper graph
shows Young’s modulus, $E$, (b) the lower graph shows Poisson’s ratio,
$\sigma$.

The least-squares analysis of the results yields that the concentration
dependence of the Young’s modulus of the gel is well described by a power law
relationship with a scaling exponent of 1.7, which is slightly smaller than
the value expected from the scaling argument deGennes . The Poisson’s ratio is
about 0.45 in the dilute gels. It decreases with the concentration of the gel
and reaches to about 0.35 at higher concentration region. The values of the
Poisson’s ratio of the dilute gels are close to those of the incompressible
materials. On the other hand, the values of the Poisson’s ratio of the dense
gels are rather close to those of the metals and the glasses. The Young’s
modulus and the Poisson’s ratio of the gel at a concentration of 1 M is
observed as $E=1.3\times 10^{4}$ N/m2 and $\sigma=0.44$, respectively. The
longitudinal modulus is then calculated as $\kappa=4.1\times 10^{4}$ N/m2 for
the gel at a concentration of 1 M. Our results are consistent with previously
published Poisson’s ratio of poly(acrylamide) gels and a theory TakigawaPJ1 ;
TakigawaPJ2 .

The diffusion coefficient of the polymer network is deduced from equation (1)
as $D_{c}=6.2\times 10^{-11}$ m2/s, which agrees with a typical value of the
diffusion coefficient of the transparent poly(acrylamide) gel evaluated by
other experiments TanakaJCP70 ; TanakaJCP59 ; Munch1 ; Munch2 .

##  4 Kinetics of water flow through gel

When the water flows in the gel (polymer network) by the application of the
mechanical pressure, the polymer network of gel deforms from the initial
position by the drag force of the water flow. We introduce a function,
$u(x,t)$, that represents the displacement of a point in the polymer network
from its initial position at $x$. We assume $u$ is always small compared to
the characteristic size of the gel, $L$. Under the present definition,
$u(x,t)=0$ at $t=0$. We also assume that the macroscopic velocity of the water
in the polymer network, $v(t)$, and microscopic friction coefficient, $f$, are
uniform in space.

$\frac{\partial v}{\partial x}=0.$ (3)

Then, we write down the equation of motion of the polymer network and that of
the water in the gel as

$\displaystyle\rho_{g}\ddot{u}$ $\displaystyle=$
$\displaystyle-f(\dot{u}-v)+\kappa\frac{\partial^{2}u}{\partial x^{2}},$ (4)
$\displaystyle\rho_{w}\dot{v}$ $\displaystyle=$ $\displaystyle
f(\dot{u}-v)-\frac{\partial P}{\partial x},$ (5)

where $\rho_{g}$ and $\rho_{w}$ are the density of the polymer network and
that of the water, respectively; A dot over the variable denotes the time
derivative; $P(x,t)$ represents the pressure inside the region that the gel
occupied.

By neglecting the inertia terms in equations (4) and (5), we obtain

$\kappa\frac{\partial^{2}u}{\partial x^{2}}-\frac{\partial P}{\partial x}=0,$
(6)

and

$f(v-\dot{u})=-\frac{\partial P}{\partial x}.$ (7)

A set of equations (3), (6), (7) corresponds to one dimensional case under a
limit of small polymer volume fraction $\phi\rightarrow 0$ of a general
description for 3 dimensional deformation of gelsYamaue ; Doi .

In our experimental setup of water flow measurement in Fig.1, we see
$P(x=0,t)=P_{0}$ and $P(x=L,t)=0$. Taking a spacial average of equation (7)
over the space the gel occupied, we obtain

$v-\frac{1}{L}\int_{0}^{L}\dot{u}dx=\frac{P_{0}}{Lf}.$ (8)

The time evolution of water flow in the gel $v(t)$ is determined locally from
$u(x,t)$ by

$v=\dot{u}-D_{c}\frac{\partial^{2}u}{\partial x^{2}}.$ (9)

Here, we use the collective diffusion coefficient of the gel, $D_{c}$, defined
in equation (1). The displacement of polymer network $u(x,t)$ is, hence,
determined by solving the equation:

$\dot{u}-\frac{1}{L}\int_{0}^{L}\dot{u}dx=D_{c}\frac{\partial^{2}u}{\partial
x^{2}}+\frac{P_{0}}{Lf}.$ (10)

In the equilibrium state, $\dot{u}(x,t)\rightarrow 0$ as $t\rightarrow\infty$.
The restoring force due to the deformation of the polymer network is in
balance with constant gradient of the pressure. In the equilibrium state,
therefore, equations (3), (6), (7), (8) give

$\frac{\partial^{2}u(x,t=\infty)}{\partial x^{2}}=-\frac{P_{0}}{D_{c}Lf}.$
(11)

The displacement, $u(x,t)$, satisfies the following boundary conditions:

$\displaystyle u(L,t)$ $\displaystyle=$ $\displaystyle 0,$ (12)
$\displaystyle\frac{\partial u(0,\infty)}{\partial x}$ $\displaystyle=$
$\displaystyle 0.$ (13)

Integration of equation (11) with respect to $x$ with the boundary conditons
yields

$u(x,\infty)=\frac{P_{0}}{2D_{c}Lf}(L^{2}-x^{2}).$ (14)

Solving equation (10) under the above boundary conditions, we obtain the
displacement, $u(x,t)$, as (See Appendix)

$u(x,t)=\sum_{n}\frac{2v_{\infty}\tau}{n^{2}}\left[(-1)^{n}\left(\cos{\frac{n\pi}{L}x}-1\right)\right]\exp{\left(-\frac{n^{2}}{\tau}t\right)},$
(15)

where

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{L^{2}f}{\pi^{2}\kappa},$ (16) $\displaystyle v_{\infty}$
$\displaystyle=$ $\displaystyle\frac{P_{0}}{Lf}.$ (17)

The velocity of the water flow, $v(t)$, is derived from equation (7) as

$v=v_{\infty}\left[1+2\sum_{n=1}^{\infty}\exp\left(-\frac{n^{2}}{\tau}t\right)\right]=v_{\infty}\theta_{3}(0,e^{t/\tau}),$
(18)

where $\theta_{3}(u,q)$ is the elliptic theta function. The position of
meniscus is expressed as

$a\left[t-2\sum_{n=1}^{\infty}\frac{\tau}{n^{2}}\exp\left(-\frac{n^{2}}{\tau}t\right)\right]+b,$
(19)

where $b$ is the initial position, and

$a=\left(\frac{R}{r}\right)^{2}v_{\infty}.$ (20)

##  5 Discussion

Now we compare the experimental results with the theoretical predictions by
equations (18) and (19). The values obtained by the experimental measurements
are, $L=1.0\times 10^{-3}$ m, $\kappa=4.1\times 10^{4}$ N/m2, and $f=6.6\times
10^{14}$ Ns/m4, respectively.

By applying these values into equation (16), we obtain the characteristic
relaxation time $\tau=1.6\times 10^{3}$ s. The equilibrium velocity of water
flow in the gel observed by the experiments is, as is already given in the
section 3, $v_{\infty}=4.4\times 10^{-9}$ m/s. From equation (14) the
deformation of gel is estimated as 3.5$\%$ which is consistent with our
assumption of small $u$.

The time evolution of the water flow is then calculated by equation (18) with
above experimental values. The results are given in Figure 5.

Figure 5: Water flow velocity in gel: (a) the solid line corresponds to the
theoretical curve fitting all the data, (b) the dotted line corresponds to the
theoretical curve expected from the short time data ($<$200s).

The normalized velocity of the water flow in the gel is also calculated from
the slope of the flow curve in Figure 3 and shown in Figure 5. We find that
the time evolution of the velocity of water flow in the gel is well explained
by the equation (18) with a single characteristic relaxation time $\tau$.

However, there is a slight discrepancy in the time evolution curve of the
velocity at a short time range ($<$200s) between the theory and the
experimental results as one can see in figure 3 and 5. The absolute value of
the equilibrium velocity determined from the short time data ($<$200s) of the
friction experiment is smaller than that determined by including the long time
data. Figure 5 shows that the deference between those two values is roughly 20
$\%$.

It would be worth noting that the theory predicts that the water flow creates
a uni-axial deformation of polymer network, i.e., the one dimensional
concentration gradient in the gel, proportional to

$\left(1+\frac{\partial u(x,\infty)}{\partial x}\right)^{-1}\approx
1+\frac{P_{0}x}{D_{c}Lf}.$ (21)

The compression of gel is the cause of the initial rapid water flow. It is
reported that the friction of the homegeneous gel depends on the polymer
volume fraction, $\phi$, as $f\propto\phi^{1.5}$ TokitaJCP95 ; deGennes . The
friction, therefore, is expected to increase according to the compression of
the gel by the water flow. The increment of the friction may explain the
difference between the short time data and long time data. The difference
between the estimated equilibrium water velocities corresponds to 14 $\%$
homogeneous compression of the sample slab gel, which is larger than the
deformation which the present theory estimates. Unfortunately, we cannot
measure the deformation of the gel under the water flow with the current
experimental setup.

##  6 Conclusion

The kinetics of the water solvent flow through the polymer network of gel is
measured experimentally and described by a simple phenomenological theory. The
experiment shows that the equilibrium state is reached after long time and
that the friction of the gel shows a large value. The time evolution of the
velocity of the water flow in the gel is calculated on the basis of the
collective diffusion model of the polymer network coupled with the water
friction, assuming both the water velocity and the friction coefficient
between the polymer network and the water are uniform in the gel. The theory
reproduces the time evolution of the water flow of the experiments for thin
dilute gel fairly well with a single characteristic relaxation time. We are
able to deduce the collective diffusion coefficient from the water flow
experiment on gels.

However there is a difference between experimental value of the equilibrium
velocity and the value expected from the theoretical calculation by using
parameters obtained from the short time flow measurements and the mechanical
measurements. It suggests that the inhomogeneity of the friction and the
concentration of the gel under the water flow might be important for the
kinetics of the water flow. A better microscopic model is required for
complete understanding of water flow through gel beyond the phenomenological
understanding presented in this paper.

###### Acknowledgements.

This work was first suggested by the late Professor Toyoichi Tanaka in 1989
during the stay of Y.Y.S. and M.T. in Massachusetts Institute of Technology.
Y.Y.S. thanks CEA, IPhT and Takushoku University, RISE for the financial
support. M.T. thanks NEDO for the financial support.

## APPENDIX A: Solution of the equation of motion

We define a function $W(x,t)$ as a deviation from the equilibrium
displacement:

$u(x,t)-\int v(t)dt\equiv W(x,t)+u(x,\infty).$ (22)

We obtain an equation for $W(x,t)$ from equation (10) as

$\frac{\partial W}{\partial t}=D_{c}\frac{\partial^{2}W}{\partial x^{2}}.$
(23)

By assuming $W(x,t)=X(x)T(t)$, the equation (23) becomes

$\displaystyle\dot{T}(t)$ $\displaystyle=$
$\displaystyle-\lambda^{2}D_{c}T(t),$ (24)
$\displaystyle\frac{d^{2}X(x)}{dx^{2}}$ $\displaystyle=$
$\displaystyle-\lambda^{2}X(x),$ (25)

where $\lambda$ is a constant. The general solution for $W(x,t)$ would be

$\displaystyle W(x,t)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\left[A_{n}\cos\lambda_{n}x+B_{n}\sin\lambda_{n}x\right]$
(26) $\displaystyle\times\exp(-\lambda_{n}^{2}D_{c}t)+A_{0}x+B_{0}.$

The boundary conditions for $W(x,t)$ are

$\displaystyle\left.\frac{du}{dx}\right|_{x=0}=0$
$\displaystyle\longrightarrow$
$\displaystyle\left.\frac{dW}{dx}\right|_{x=0}=0,$ (27)
$\displaystyle\left.\frac{du}{dt}\right|_{t=\infty}=0$
$\displaystyle\longrightarrow$
$\displaystyle\left.\frac{dW}{dt}\right|_{t=\infty}=v(\infty),$ (28)
$\displaystyle u(L,t)=0$ $\displaystyle\longrightarrow$ $\displaystyle
W(L,t)=-\int vdt.$ (29)

The boundary condition (27) yields

$\displaystyle\left.\frac{dW}{dx}\right|_{x=0}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}[A_{n}(-\lambda_{n}\sin\lambda_{n}x)+B_{n}(\lambda_{n}\cos\lambda_{n}x)]$
$\displaystyle\left.\times\exp(-\lambda_{n}^{2}D_{c}t)\right|_{x=0}+B_{0}$
$\displaystyle=$
$\displaystyle\left.\sum_{n=1}^{\infty}[A_{n}(-\lambda_{n}\times
0)+B_{n}(\lambda_{n}\times 1)]\exp(-\lambda_{n}^{2}D_{c}t)\right|_{x=0}$
$\displaystyle+B_{0}$ $\displaystyle=$ $\displaystyle 0.$

This indicates $B_{n}=0$.

The boundary condition (28) yields

$\displaystyle\left.\frac{dW}{dx}\right|_{t=\infty}$ $\displaystyle=$
$\displaystyle\left.\sum_{n=1}^{\infty}[A_{n}\cos\lambda_{n}x](-\lambda_{n}^{2}D_{c})\exp(-\lambda_{n}^{2}D_{c}t)\right|_{t=\infty}+A_{0}$
$\displaystyle=$ $\displaystyle v(\infty).$

This indicates that the constant, $\lambda_{n}^{2}$, should be positive,
$\lambda_{n}^{2}>0$ for $n\geq 1$ and $A_{0}=v(\infty)$. Finally, the boundary
condition (29) yields

$\displaystyle W(L,t)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}[A_{n}(\cos\lambda_{n}L+C_{nA})]\exp(-\lambda_{n}^{2}D_{c}t)+A_{0}L+B_{0}$
$\displaystyle=$ $\displaystyle-\int vdt.$

This indicates $C_{nA}=-\cos\lambda_{n}L$. The following relationships for
$B_{n}$, $\lambda_{n}^{2}$, and $C_{nA}$ are obtained:

$\displaystyle B_{n}$ $\displaystyle=$ $\displaystyle 0,$ (30)
$\displaystyle\lambda_{n}^{2}$ $\displaystyle>$ $\displaystyle 0,$ (31)
$\displaystyle C_{nA}$ $\displaystyle=$ $\displaystyle-\cos\lambda_{n}L.$ (32)

And $W(x,t)$ becomes

$W(x,t)=\sum_{n=1}^{\infty}[A_{n}(\cos\lambda_{n}x+C_{nA})]\exp(-\lambda_{n}^{2}D_{c}t).$
(33)

Substitution of this solution (33) into equation (23) yields

$\displaystyle\sum_{n=1}^{\infty}[A_{n}(\cos\lambda_{n}x+C_{nA})](-\lambda_{n}^{2}D_{c})\exp(-\lambda_{n}^{2}D_{c}t)$
$\displaystyle-\frac{1}{L}\int_{0}^{L}\sum_{n=1}^{\infty}[A_{n}(\cos\lambda_{n}x+C_{nA})]$
$\displaystyle\times(-\lambda_{n}^{2}D_{c})\exp(-\lambda_{n}^{2}D_{c}t)dx$
$\displaystyle=D_{c}\sum_{n=1}^{\infty}A_{n}\lambda_{n}^{2}(-\cos\lambda_{n}x)\exp(-\lambda_{n}^{2}D_{c}t).$
(34)

The first term of the left handed member (LHM1) is written as follows:

$\displaystyle{\rm LHM1}=$
$\displaystyle\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})A_{n}(\cos\lambda_{n}x)\exp(-\lambda_{n}^{2}D_{c}t)$
$\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})(A_{n}C_{nA})\exp(-\lambda_{n}^{2}D_{c}t).$

The second term of the left handed member (LHM2) is expressed as follows:

$\displaystyle{\rm LHM2}=$ $\displaystyle-$
$\displaystyle\frac{1}{L}\int_{0}^{L}\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})(A_{n}\cos\lambda_{n}x)$
$\displaystyle\times\exp(-\lambda_{n}^{2}D_{c}t)dx$ $\displaystyle-$
$\displaystyle\frac{1}{L}\int_{0}^{L}\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})(A_{n}C_{nA})\exp(-\lambda_{n}^{2}D_{c}t)dx.$

The first term of LHM2 is expressed as

$\displaystyle-$
$\displaystyle\frac{1}{L}\int_{0}^{L}\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})(A_{n}\cos\lambda_{n}x)\exp(-\lambda_{n}^{2}D_{c}t)dx$
$\displaystyle=$ $\displaystyle-$
$\displaystyle\left.\frac{1}{L}\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})\frac{1}{\lambda_{n}}(A_{n}\sin\lambda_{n}x)\exp(-\lambda_{n}^{2}D_{c}t)\right|_{0}^{L}$
$\displaystyle=$ $\displaystyle-$
$\displaystyle\frac{1}{L}\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})\frac{1}{\lambda_{n}}(A_{n}\sin\lambda_{n}L)\exp(-\lambda_{n}^{2}D_{c}t).$

The second term of LHM2 becomes

$\displaystyle-$
$\displaystyle\frac{1}{L}\int_{0}^{L}\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})(A_{n}C_{nA})\exp(-\lambda_{n}^{2}D_{c}t)dx$
$\displaystyle=$ $\displaystyle-$
$\displaystyle\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})(A_{n}C_{nA})\exp(-\lambda_{n}^{2}D_{c}t).$

Hence, the equation (34) is reduced to

$-\frac{1}{L}\sum_{n=1}^{\infty}(-\lambda_{n}^{2}D_{c})A_{n}\frac{1}{\lambda_{n}}(\sin\lambda_{n}L)\exp(-\lambda_{n}^{2}D_{c}t)=0.$

This indicates

$\lambda_{n}=\frac{n\pi}{L},$ (35)

and

$C_{nA}=-\cos(n\pi)=-(-1)^{n}.$ (36)

The solution, $W(x,t)$, can be now written as

$W(x,t)=\sum_{n=1}^{\infty}A_{n}\left(\cos\frac{n\pi}{L}x-(-1)^{n}\right)\exp\left(-\frac{n^{2}\pi^{2}}{L^{2}}D_{c}t\right).$
(37)

Since the initial condition is

$W(x,0)=\frac{P_{0}}{2D_{c}Lf}(x^{2}-L^{2}),$ (38)

the coefficients, $A_{n}$, are determined by the Fourier series expansion:

$\sum_{n=1}^{\infty}A_{n}\left(\cos\frac{n\pi}{L}x-(-1)^{n}\right)=\frac{P_{0}}{2D_{c}Lf}(x^{2}-L^{2}).$
(39)

We find

$A_{n}=\frac{(-1)^{n}}{n^{2}\pi^{2}}\frac{2P_{0}L}{D_{c}f}.$ (40)

The displacement, $u(x,t)$, is now determined as

$u(x,t)=\sum_{n=1}^{\infty}\mathcal{A}_{n}\left[(-1)^{n}(\cos
q_{n}x-1)\right]\exp(-\Gamma_{n}t),$ (41)

where, $\mathcal{A}_{n}$, $q_{n}$, and $\Gamma_{n}$ are given by

$\displaystyle\mathcal{A}_{n}$ $\displaystyle=$
$\displaystyle\frac{2P_{0}L}{n^{2}\pi^{2}D_{c}f},$ (42) $\displaystyle q_{n}$
$\displaystyle=$ $\displaystyle\frac{n\pi}{L},$ (43) $\displaystyle\Gamma_{n}$
$\displaystyle=$ $\displaystyle\left(\frac{n\pi}{L}\right)^{2}D_{c}.$ (44)

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