신소재공학

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surface (which is assumed to remain constant).
(6) Error Function (erf)
The error function (erf) is a mathematical function that can be read from a table or a graph to get the value of the error function for any argument. Using a dummy variable Z, the error function provides one solution for Fick's second law:
Where Z is a variable of convenience. Mathematically, the error function is the integral of the Gaussian curve, but it is necessary to understand the mathematics of the error function in order to use it. You can treat it like the sine function, log function, or anything else, and learn to look up values to go back and forth between Z and erf(Z) on a table or graph.
(7) ERF function graph
The error function can be read from either a table or a graph. In a given problem, you will either have the Z value and need to look up erf(Z), or you will have the erf(Z) value and need to look up the corresponding Z value. In either case, simply read it off of the graph or table given.
ERF function table
z erf (z) z erf (z) z erf (z)
0 0 0.55 0.5633 1.3 0.9340
0.025 0.0282 0.60 0.6039 1.4 0.9523
0.05 0.0564 0.65 0.6420 1.5 0.9661
0.10 0.1125 0.70 0.6778 1.6 0.9763
0.15 0.1680 0.75 0.7112 1.7 0.9838
0.20 0.2227 0.80 0.7421 1.8 0.9891
0.25 0.2763 0.85 0.7707 1.9 0.9928
0.30 0.3286 0.90 0.7970 2.0 0.9953
0.35 0.3794 0.95 0.8209 2.2 0.9981
0.40 0.4284 1.0 0.8427 2.4 0.9993
0.45 0.4755 1.1 0.8802 2.6 0.9998
0.50 0.5205 1.2 0.9103 2.8 0.9999
(8) Carburizing
Carburization is a classic example of using diffusion in the processing of materials. It is the process by which carbon atoms are diffused into the surface of a steel, in order to harden it. This process is frequently employed for such things as tools and gears that require a tough and ductile interior but a hard contact surface.
There are several different types of problems in this chapter that have to do with carburizing. Each is worked out in detail using Mathcad, to give you some practice solving for different variables, as well as to practice using the error function. Note that a problem may ask you to solve for
a) the depth at which a particular concentration exists;
b) the concentration at a given depth;
c) the time required to achieve a given depth or concentration; or
d) the temperature required to accomplish this depth or concentration in a given time.
(9) Carburizing Example
On this graph, you can see that Co stands for the original bulk concentration,
Cx is the concentration at a depth X, and Cs is the concentration of the diffusing
atoms at the surface (which is assumed to remain constant).
(10) Carburizing Problems
These problems can be worked with MathCad to better understand the calculations involving Fick's second law as it applies to carburization, and to develop some facility with the mathematics. Example Problems 12, 16 and 17 calculate the time required. Example Problems 13 and 14 calculate the carbon concentration, and Example Problems 15 and 18 calculate the temperature required.

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신소재공학,   원자결합,   결정구조,   결함,   확산