TRANSPORT PROCESSES

THEORETICAL
	Both diffusion and heat conduction belong to the family of conductive transport processes: transference of extensivity (number of moles and internal energy, respectively) is effected by random walk (Brownian motion) of carrier particles. Carriers in diffusion are the mobile molecules and ions while in heat conduction they are free electrons and phonons. (Phonons are quasi-particles related to lattice vibrations of solid bodies. As we measure insulators, the only carriers are phonons in our case.)
	Phenomena like these call for statistical mechanics. However, they can be described phenomenologically as well, analougously to the best known transport process electric current. Some compromises, theoretically embarrassing but practically not too significant, are to be made along this way.

1.	Phenomenological description
1.1.	Transport equation
	Let us begin the treatment with the simplest case. Under homogeneous and stationary conditions the electric current obeys Ohm’s law:
										1)
or:
										2)
, that is electric current density (flux) is proportional to electric field strength V/X.
	On the same conditions, the analougous Ohmian equations for our conductive transport processes are:

									3)

The flux of an extensivity is proportional to the “field strength” of its conjugated intensivity. The negative sign in Equations 3) preserving positivity of transport coefficients  and λ is absent in the original Ohm’s law, due to conventional direction of current and voltage. An other difference is more serious: electric voltage is directly measurable while its analogue “chemical voltage”  is not and its relation to the measurable concentration is nonlinear however monotonic:

								4)

Due to this monotonic character, if a concentration dependent transport coefficient is accepted, Equation 3) for Jn may be rewritten with concentration: 

										5)

	If the space is not homogeneous the above relations may hold true of an infinitesimally small legth dX:

 								6)

and if it has more than one dimensions the above equations describe the transport process in each direction:

													     7)

 and  are three-dimensional vectors:

					8)

where   és  are the unit vectors of the space. Application of differential operator gradient, in Cartesian coordinates

							9)

gives Fick’s I. law and the law of heat conduction:

							7)

	In order to extend the Ohmic description to non-stationary processes one has to take into account the related continuity equation. It must be valid for any infinitesimally small (dX, dY, dZ, dt) space-time domain as it is a conservation law with very simple scheme:

				   inflow of extensivity into the domain
			-	outflow of extensivity from the domain
			      increase of extensivity inside of the domain

In dimension X it takes the form:

10)

, where   is the surface perpendicular to direction X,  the volume, ρ the density and the specific heat of the domain. As the expressions in brackets are the negative differentials of   and , respectively, Equations 10) can be broght to more concise form:

							11)

Clearly, one has to sum up the effects of each vector component in a three-dimensional space:

 				   12)

 Making use of an other differential operator divergence what is in Cartesian coordinates: 

			,				13)

Equations 12) can be written in more general and concise form:

								14)

One can insert flux vectors Jn and JU from Equation 7) :

					15)

and gets the most general forms of  Fick’ II and  and Fourier’s laws.  In order to avoid nonlinear partial differential equations, what Equations 15) essentially are, one has to regard diffusivity D independent of concentration c and heat conductivity λ independent of temperature T. Doing this, they become independent of space variables, too, so they are constants:

					16)

, where  is  the Laplacian, in Cartesian coordinates:

			,					17)

and a is the thermal diffusivity: 
										18)
	Based on their geometrical meaning, Equations 16) are called differential equations of smoothing. As the second derivative of a function gives its curvature (the reciprocal of the radius of the tangent circle), the geometrical meaning of Equations 16) is the following:
	Rate of intensivity change is proportional and of the same sign to its curvature at any place. 
	Consequently, except for the initial state, the concentration – space or the temperature – space function  must be continuous and differentiable at any point. The final state must be one where these functions are everywhere the possibly smoothest. (A straight line in Cartesian coordinates, however, in other coordinates it may be different!) 

1.2.	Transport coefficients
	It was shown above that diffusivity D depended on concentration even if the conductivity k of the Ohmic description did not. Luckily enough, this concentration dependence is usually not steep, so it is tolerable. 
	The dependence on temperature is more serious. It is easy to understand as each transport coefficient is a product of three factors:

										19)

, where x is the fraction of mobile particles, l is their mean free path and v is their average speed.
	For a gas at constant pressure x=1independently of temperature,  and , consequently:

										20)

For a liquid or a solid, temperature dependence of  l  is negligible, , but the volume fraction of vacancies, consequently  x obeys the Arrhénius law and the latter rules the temperature dependence of  diffusivity D.
	Thermal diffusivity a of an insulator solid body grows with temperature steeply at low temperatures while at higher ones decreases exponentially, at even higher ones hyperbolically as number of phonons strongly grows with temperature but the phonons are scattered both on vacancies and on each other.

1.3.	Conditions for uniquness of solution
	Equations 16) are linear homogeneous second order partial fifferential equations. Solution of any differential equation needs integration. Result of an integration is unique only if an integration constant is given. “Partial integration constants” are needed for a partial differential equation: for instance, if a time integral is calculated, the integration constant may not depend on time but may be a function of the rest of variables. Partial integration constant for the time integral is called initial condition while that for a space variable integral is called boundary condition. Of course, two boundary conditions are needed for any space variable. (Some authors do not distinguish initial conditions, they call them boundary conditions, too.) 
	 So, to get a c(t,X) function as solution one needs an initial condition of form
 
			c(0,X)=...							21)

and two boundary conditions of form

			      c(t,X1) =...							22a)
	and/or	 	=... 						22b)

	A symmetry can replace a boundary condition as, together with the geometrical meaning  of  Equations 16), it provides for a boundary condition.  For instance, let concentration have an axis of symmetry at place X°. Being symmetrical and having finite curvature  at the same time are possible only if the function satisfies the condition 

									23)

Before writing a boundary condition one has to compare the length available for the transport process to the path  made by random walk in time of the experiment. If the latter is shorter than the first, it is more appropriate mathematically to regard the space infinite and write the bondary condition at infinity.  Maximum of the path made by random walk in time tm :

				and  , respectively		24)

 Equation 24) gives the treble of the standard deviation  of a Gaussian curve. Probability density function of a continuous stochastic variable is a Gaussian curve if it is effected by many independent elementary steps.  99,7% of the integral of the Gaussian, that is 99,7% of the mobile particlesss, is in its  3 section:  99,7% of the mobile particles is not farer from its initial position than 3. This is a theoretical maximum as Gaussian only tends to 0 at 3 but the real probability density function is exactly 0 at those places. Were it according to Gaussian, there would be some particles that could move to infinity in finite time, that is they would have infinite speed. This is the infinite speed paradox of the Ohmian description.

	Clearly, any transport process is of infinite space in the beginning and later, if enough time is allowed to it, becomes to one of finite space.

2. Diffusion of concentration-jump in a one-dimensional infinite space 
	Let the left half of a tube be filled with a solution of homogeneous solute concentration c and the right half with clean solvent. Be the length of both halves considerably longer than the path given by Equation 24). Then, diffusion is of one dimension (along axis of the tube) and of infinite space. Be the origin of the coordinates at the half of the tube. 
	The complete mathematical problem is as follows:

	unknown function:			c = c(t,X)				25)
	related differential equation:					26)
			initial coondition:	c(0,X) =0  if X≥0;	c  if X<0	27)
			boundary conditions:	c(t,) = 0				28a)
						c(t,-) = c				28b)

	Omitting the details of solution, the final result is:

						29)

This is an error function (exactly, a complementary error function), integral of a Gaussian curve. One can note the peculiarity of the transport processes in infinite space: time dependence is of square root.  

3. Heat conduction of temperature-jump in a three-dimensional finite space
	Let us be given a brick-shaped body with edges in length 2X0, 2Y0, 2Z0 and of homogeneous temperature T0 . Now, let us take over this body into a new environment of infinite heat capacity and temperature Tr  (Tr   T0). Let the origin of the coordinates be set into the center and the axes perpendicular to the faces of the body (Figure 1).  

						Figure 1

As temperature-space functions must be clearly symmetrical to the axes, commentary to Equation 23) applies to them at the origin. 
	The complete mathematical problem:

	unknown function:			T = T(t,X,Y, Z)			    30)

	related differential equation:		    	    31)

	initial condition: T(0,X,Y,Z) = T, if -X0<X< X0 and -Y0<Y< Y0 and -Z0<Z< Z0;		   		                = Tr   in any other case				    32)

	boundary conditions:   T(t, ±X0,Y,Z) = Tr , if -Y0≤Y≤Y0 and -Z0≤Z≤Z0		
			             T(t, X,±Y0,Z) = Tr , if -X0≤X≤X0 and -Z0≤Z≤Z0	   33a)
			        	T(t, X,Y,±Z0) = Tr , if -X0≤X≤X0 and -Y0≤Y≤Y0				    
     , if -Y0<Y<Y0 and -Z0<Z< Z0
     , if -X0<X<X0 and -Z0<Z< Z0		  33b)
			     , if -X0<X<X0 and -Y0<Y< Y0

	Solution of this problem, omitting the mathematical details again, is as follows:

			where 			    34)

There is a remarkable difference from transport processes of infinite space: time dependence is exponential in finite spaces. However, there is a common feature, too: time needed to achieve a given concentration or temperature distribution is proportional to the square of the size.

EQUIPMENT 
Diffusivity measurement
According to Figure 2, a simple equipment is assembled to measure the diffusivity of a salt/water system.

					Figure 2

The equipment is composed of an optical rail (1), a pointer laser (2), a large cuvette (3) and a screen (4, a paper sheet on the wall). The laser is to be aligned such that its light beam reach the cuvette at the altitude of maximum concentration gradient.  If the direction of gradient is down, the upper part of the light beam, being in a liquid of smaller index of refraction, proceeds faster than the lower part. (Velocity of light and, consequently, the distance made by it are inversely proportional to index of refraction.) For instance, let us be given a light beam of width dx. Its uppermost part proceeding in a medium of index of refraction n-dn has done the optical length of the cuvette l while its downmost part proceeding in a medium of index of refraction n has done merely l-dl:

		after simplified:				35)

Due to this path-difference, light beam is bent from normal direction downward and arrives at the opposite wall of the cuvette with angle α. 

										36)

When it enters air it is refracted again, proceeds with angle β and arrives at the screen at distance L in position z instead of 0.  (The tiny shift in the wall of the cuvette is negligated.)

										37)

Inserting dl to Equation 36) from Equation 35), then writing the Snellius law of refraction with Equations 36) and 37) (approximating sines by tangents, due to the small angles)  gives the relation between departure z and the gradient of index of refraction: 

										38)

There is a linear relation between index of refraction and concentration:

		, consequently			39)

A refractometer belongs to this measurement to measure indeces of refraction of distilled water and the salt solution. 
Order of magnitude of diffusivities of small molecules in liquid phase at room temperature is 10-9 m2/s, time of the measurement is 15-20 minutes (ca. 103 s), so the maximum path of random walk according to Equation 24) must be in order of magnitude of  mm’s. As both water and the salt solution have height of several cm’s, diffusion taking place is of infinite space described by Equation  29). 
As departure z is proportional to the gradient of index of refraction that is to that of concentration, Equation 29) has to be derived:

 							40)

The maximum of concentration gradient is at X=0 :

									41)

Taking concentration and its gradient from Equation 39) and gradient of index of refraction from Equation 38) one can reformulate Equation 41) to be ready for evaluation of this measurement:

							42)

There is a slide-gauge and a tape measure for optical length l and screen distance L measurements. An ordinary watch is perfect for measuring elapsed time.

Thermal diffusivity measurement
A brick-shaped polyamide body is immersed into an icy bath being in a Dewar vessel (Tr.= 0˚C). The body has a thermocouple, one of its junctions is in the center while the other one on the surface, consequently, its voltage is proportional with temperature of the center expressed in ˚C. This voltage is amplified by a home-made amplifier, measured by a digital multimeter and archived in a txt-file on a computer.
Order of magnitude of thermal diffusivities of insulating solid materials at room temperature is 10-7 m2/s, time of the measurement is 15-20 minutes (ca. 103 s), so the maximum path of random walk according to Equation 24) must be in order of magnitude of cm’s. Distances of the surfaces from the center are 1.5 – 5 cm, so at least for longer measuring times, thermal diffusion can be regarded as one of finite space described by Equation 34). 
The temperature of the geometrical center can be got from Equation 34) by inserting X=Y=Z=0 and Tr=0 :

				43)

As, according to Equation 34), value of a time constant τi steeply decreases with increasing index i, merely the term of i=0 will be the only non-negligible one in the sum of Equation 43) after a transient period:

			,    if t>ttransient				44)

After the transient period logarithm of the temperature of the center must decrease with time linearly and the slope of this straight line is -1/τ0.
	Length of edges can be measured with a slide-gauge.

PROCEDURE AND CALCULATION
Diffusivity measurement
1.	Stand the optical rail perpendicularly to the wall and be its cuvette at a distance 0.7- 1m from it. Pipette 50cm3 distilled water into the cuvette.
2.	Switch on the laser by pulling the O-ring to the switch. Set the position of the laser and/or that of the cuvette such that the light beam cross the cuvette at the water surface.
3.	Attach a longer paper sheet to the wall where the light is thrown. 
4.	Pipette gently 50cm3 salt solution through the funnel of the cuvette to sublayer the solution to water. The light beam has to cross the cuvette at the interface of the two liquids at the end of sublayering. If it does not, only the position of the laser may be changed!
5.	Mark the position of the light point on the screen and begin measuring time. Repeat marking regularly with a 1-3 minute period. Meantime indices of refraction can be measured. (That of air can be taken 1.000.)
6.	When at least 8 points have been measured, measure screen distance L, then remove the cuvette from the optical rail gently and mark the position of the base point on the screen. 
7.	Switch off the laser, clean the cuvette and measure its optical length l. Gently remove the paper sheet from the wall.
8.	Measure values of z’s: distances of the marked points from the base point. Make a time versus 1/z2 plot and, making use of Equation 42), calculate diffusivity from its slope.

Thermal diffusivity measurement
1.	Short circuite the input of the amplifier, set the data logging method “repetitive” and draw a base line. 
2.	Measure the edges of the plastic body with the slide-gauge.
3.	Connect the cables of the thermocouple to the amplifier input. Check the sign of voltage by touching the junction on the surface; the sign during the measurement will be opposite. Change the two cables if needed. Without disconnecting cables, short circuite the input again.
4.	Take ice cubes from the refrigerator, pack them in a larger piece of cloth and break it to powder with a wooden hammer (on the curb, not on the table!). Take the ice powder to the Dewar vessel, and add some water. 
5.	Push the plastic body into the icy bath and add more ice and water if it is not covered completely. Remove short circuit and register the amplified thermovoltage for 12-20 min. Then short circuite again and draw base line again.
6.	Make a logarithm of base line corrected signal versus time plot. Fit a straight line for the points which clearly not belong to the transient period and read its slope.
7.	Write i=0 into Equation 34) and calculate thermal diffusivity with it from the slope.