In conclusion, we have used HJE method to get the solution of modified Eigen model in Hamming class. Based on the two symmetrical assumptions, the symmetrical distri-bution and fitness function, the modified Eigen model becomes a solvable problem for large genome length. In this approach, the steady-state probability of Hamming class with O(1/N ) relative accuracy has been obtained by working carefully with the combinatorial problems during the calculation process, where N is the genome length. These calcula-tions are much harder than the corresponding calculacalcula-tions for the Crow-Kimura case [51].
The properties for the O(N1) correction terms have been investigated completely in these calculations. The analytical results for the steady-state probability of Hamming class is well consistent with the numerical results simulated by Runge-Kutta method, where the relative errors between the analytical and numerical results are less than 1 % as shown in Figs. 4.19-21 and table 4.1-3. Our formula for the O(N1) correction of probability in Hamming class, Eq. (4.64), is not the special case, and one can apply this formula for any symmetrical distribution and fitness function. Furthermore, our results can be applied to get accurate expression for the steady-state probability of Hamming class for the case of large genome length, where the numerics cannot give the accurate results. In our model we work with the mutation parameter γ = N (1−q), while in [52] the mutation parameter µ = N (1−q)/q has been considered. Our expressions for the corrections of zero and first order in N1 for the mean fitness is identical to the results derived in [52] by quantum field theory.
5.3 SMAT Modelling
Experimentally, it is observed that the grain size is appreciably refined by SMAT, from the initially about 20 µm down to less than 100 nm, as shown in Fig. 5.1(a) with a gradient trend as viewed from the sample cross-section [47]. In parallel, the hardness increases from the initial about 2.7 GP a up to about 6.0 GP a, as shown in Fig. 5.1(b) [47]. The kinetic energy from the flying balls appear to effectively induce substantial internal or
strain energy into the sample surface, increasing the dislocations and other defects, refining the grain size, and raising the hardness. The ball speed can be estimated from Eq. (3.59) to be within the range of 5 ∼ 10 m/s, and the kinetic energy for all the flying balls can also be estimated to be about 10 ∼ 120 mJ. From the experimental results in Fig. 5.1, coupled with the estimated values based on the current analytical model, it appears that the optimum speed for the 304 stainless steel might be around 8∼ 10 m/s and the optimum kinetic energy might be around 70 ∼ 75 mJ. The adjustment of the SMAT parameters will influence accordingly the speed (in Fig. 3.1), kinetic energy (in Fig. 3.2), flying time period (in Fig. 3.3), power (in Fig. 3.4), and temperature profiles within the experienced range of the samples (in Figs. 4.22-23).
In this model, we have made efforts in evaluating the temperature profile from the bombarded surface to the sample inner portion (Figs. 4.22-23). This profile can be used as a reference in assessing the experienced temperature at the particular sample depth. For example, based on the calculated temperature in Fig. 4.23 for the 304 stainless steel, the temperature at the depth of 200 µm from the surface would be 365 K or 92.
The other parameter left would be the strain rate. In accordance with Eq. (4.78), the strain rate would vary from 3×102 ∼ 5×102s−1. Taking the 4×102s−1as the mean value, and 92 as the experienced temperature, we can incorporate into Eq. (4.74) to extract the Zener-Holloman Z parameter, which is useful for estimate the materials microstructure properties. For 304 stainless steel, the governing activation energy Q should be related to the Fe diffusion, and Q ∼ 220 kJ/mol is a logical value [53, 54]. With the above information and the gas constant R = 8.3 J/K, Z can be calculated to be 1.4× 1034s−1. With the same calculation manner, we can estimate all values for various cross-sectional positions of the SMAT sample, and plot the measured grain size and Zener-Holloman Z parameter, as presented in Fig. 5.2.
Thus, for SMAT researchers, we can first design the SMAT working parameters (based on the needs), and can calculate the resulting speed, temperature, strain rate, and energy based on this model in Figs. 3.1-4 and Figs. 4.22-23. With all the information, we can es-timate the grain size from the Zener-Holloman Z parameter based on Fig. 5.2. The current
approach and modelling nicely establish the link between the physics and the engineering material surface modifications.
Appendix
.1 The Coefficient of Finite Difference
The following tables present the coefficients of the forward finite difference with space h for several order accuracy in h.
Table 1: The coefficient table for the forward finite difference of f′(x).
Accuracy f (x) f (x + h) f (x + 2h) f (x + 3h) f (x + 4h) f (x + 5h)
O(h) −1 1
O(h2) −3/2 2 −1/2
O(h3) −11/6 3 −3/2 1/3
O(h4) −25/12 4 −3 4/3 −1/4
O(h5) −137/60 5 −5 10/3 −5/4 1/5
Table 2: The coefficient table for the forward finite difference of f′′(x).
Accuracy f (x) f (x + h) f (x + 2h) f (x + 3h) f (x + 4h) f (x + 5h)
O(h) 1 −2 1
O(h2) 2 −5 4 −1
O(h3) 35/12 −26/3 19/2 −14/3 11/12
O(h4) 15/4 −77/6 107/6 −13 61/12 −5/6
For example, the first derivative of f (x) with O(h3) accuracy is:
f′(x)≈ −116 f (x) + 3f (x + h)− 32f (x + 2h) + 13f (x + 3h)
h ,
and the second derivative of f (x) with O(h2) accuracy is:
f′′(x)≈ 2f (x)− 5f(x + h) + 4f(x + 2h) − f(x + 3h)
h2 .
.2 The Power Series Expansion of Kummer’s Function
The expansion in power series for KummerM (a, b, x) with a > 0 and b > 0 near x = 0:
KummerM (a, b, x)
= 1 + a
bx + a(a + 1)
2b(b + 1)x2+ a(a + 1)(a + 2) 6b(b + 1)(b + 2)x3 + a(a + 1)(a + 2)(a + 3)
24b(b + 1)(b + 2)(b + 3)x4+ a(a + 1)(a + 2)(a + 3)(a + 4)
120b(b + 1)(b + 2)(b + 3)(b + 4)x5+ O(x6).
The expansion in power series for KummerU (a, b, x) with b̸= Z near x = 0:
KummerU (a, b, x) = Γ(1− b)
Γ(a− b + 1)[1 + a
bx + a(a + 1)
2b(b + 1)x2+· · · ] + Γ(b− 1)
Γ(a) x1−b[1 + 1 + a− b
2− b x + (1 + a− b)(2 + a − b)
2(2− b)(3 − b) x3+· · · ].
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Figures and Tables
Figure 1.1: (a) The desiged-dimension of chamber in the SMAT experiment. (b) The schematic drawing showing that the sample material gains the heat and strain energy from the kinetic energy loss of sample and flying balls.
Figure 2.1: The Schematic of sample configuration.
Figure 3.1: (a) The average speed of flying balls (in Eq. (3.59)) versus the SMAT ampli-tude for the parameters, H = 20 mm, D = 3 mm, ω = 40π krad/s, and mmb
m = 10−6. (b) The average speed of flying balls (in Eq. (3.59)) versus the SMAT angular frequency for the parameters, H = 20 mm, D = 3 mm, A = 60 µm, and mmb
m = 10−6.
Figure 3.2: (a) The variation trend of ∆Ek,loss,bpredicted by Eq. (3.60) as a function ofmmb
s
for the parameters H = 20 mm, A = 60 µm, ω = 40π krad/s,mmb
m = 10−6, and e = 0.25, (b) the variation trend of ∆Ek,loss,s predicted by Eq. (3.61) as a function of mmb
s for the parameters H = 20 mm, A = 60 µm, ω = 40π krad/s,mmb
m = 10−6, and e = 0.25, (c) the variation trend of total energy loss, i.e., the sum of ∆Ek,loss,b and ∆Ek,loss,sas a function of mmb
s for the parameters H = 20 mm, A = 60 µm, ω = 40π krad/s, mmb
m = 10−6, and e = 0.25.
Figure 3.3: (a) The averaged time period of flying balls predicted by Eq. (3.64) versus the SMAT amplitude for the parameters H = 20 mm, ω = 40π krad/s, mmb
m = 10−6, and e = 0.25. (b) The averaged time period of flying balls predicted by Eq. (3.64) versus the SMAT angular frequency for the parameters H = 20 mm, A = 60 µm, mmb
m = 10−6, and e = 0.25.
Figure 3.4: (a) The variation trend of Ploss,b predicted by Eq. (3.65) as a function of mmb
s
for the parameters H = 20 mm, A = 60 µm, ω = 40π krad/s, mmb
m = 10−6, and e = 0.25. (b) The variation trend of Ploss,spredicted by Eq. (3.66) as a function of mmb
s for the parameters H = 20 mm, A = 60 µm, ω = 40π krad/s,mmb
m = 10−6, and e = 0.25. (c) The variation trend of the total power loss, i.e., the sum of Ploss,band Ploss,s as a function of mmb
s for the parameters H = 20 mm, A = 60 µm, ω = 40π krad/s, mmb
m = 10−6, and e = 0.25.
Figure 4.1: The mechanism for DNA-mRNA-protein process.
Figure 4.2: The transition PDF for mRNA-protein process.
0 0.5 1 1.5 2 0
1 2 3
x (nM) t = 14 s
t = 1400 s t = 2800 s t = 4200 s P(x,t)
Figure 4.3: The simulation for the dynamical state of PDF with parameters: a = 0.5 and b = 5 from t = 14 s∼ 4200 s.
0 0.5 1 1.5
0 1 2 3
x (nM) t = 5600 s
t = 9800 s P(x,t)
Figure 4.4: The simulation for the dynamical state of PDF with parameters: a = 0.5 and b = 5 from t = 5600 s∼ 9800 s.
0 0.2 0.4 0.6 0.8 1 0
1 2 3
x (nM) t = 25200 s
Analytical Sol.
P(x,t)
Figure 4.5: The simulation at t = 25200 s for the dynamical state of PDF and analytical solution with parameters: a = 0.5 and b = 5.
0 2 4 6 8
0.2 0.4 0.6 0.8
x (nM) t = 8 s
t = 200 s t = 400 s t = 600 s t = 800 s t = 1000 s P(x,t)
Figure 4.6: The simulation from t = 8 s ∼ 1000 s for the dynamical state of PDF with parameters: a = 5 and b = 5.
0 20 40 60 0
0.05 0.1 0.15
x (nM) t = 1200 s
t = 1400 s t = 1600 s t = 2000 s
Analytical Sol.
P(x,t)
Figure 4.7: The simulation from t = 1200 s ∼ 2000 s for the dynamical state of PDF and analytical solution with parameters: a = 5 and b = 5.
0 20 40 60
0 0.01 0.02 0.03 0.04
x (nM) t = 2400 s
t = 4800 s t = 7200 s t = 12000 s Analytical Sol.
P(x,t)
Figure 4.8: The simulation from t = 2400 s∼ 12000 s for the dynamical state of PDF and analytical solution with parameters: a = 5 and b = 5.
0 5 10 15 20 0.2
0.4 0.6 0.8
x (nM) t = 40 s
t = 200 s t = 400 s t = 600 s t = 1000 s P(x,t)
Figure 4.9: The simulation from t = 40 s∼ 1000 s for the dynamical state of PDF with parameters: a = 8 and b = 8.
0 50 100
0 0.005 0.01 0.015 0.02
x (nM) t = 1600 s
t = 2800 s t = 4400 s t = 6800 s t = 9600 s
Analytical Sol.
P(x,t)
Figure 4.10: The simulation from t = 1600 s∼ 9600 s for the dynamical state of PDF and analytical solution with parameters: a = 8 and b = 8.
0 2 4 6 8 10 0
0.1 0.2 0.3
ε =0.1 ε =1 ε =10 P(x)
x
Figure 4.11: The steady-state of PDF for Eq. (4.11) with parameters: a = 2 , k = 1, and γ2 = 1.
0 2 4 6 8
0 0.5 1
1.5 k=0.5
k=1 k=10 P(x)
x
Figure 4.12: The steady-state of PDF for Eq. (4.11) with parameters: a = 2, ϵ = 0.1, and γ2 = 1.
0 0.5 1 1.5 2 0
2 4 6
t = 0 t = 640 t = 1120 t = 4480
Steady State
x P(x)
Figure 4.13: The simulation from t = 0 ∼ 4480 for the dynamical state of PDF and analytical solution with parameters: a = 0.5, ϵ = 2× 10−6, γ2 = 2× 10−3, and k = 1.
0 1 2 3
0 0.5 1
1.5 t = 0
t = 200 t = 1200
Steady State
x P(x)
Figure 4.14: The simulation from t = 0 ∼ 1200 for the dynamical state of PDF and analytical solution with parameters: a = 0.5, ϵ = 2× 10−4, γ2 = 2× 10−3, and k = 1.
0 2 4 6 0
0.5 1
1.5 t = 0
t = 240 t = 600 t = 4320
Steady State
x P(x)
Figure 4.15: The simulation from t = 0 ∼ 4320 for the dynamical state of PDF and analytical solution with parameters: a = 2, ϵ = 2× 10−6, γ2 = 2× 10−3, and k = 1.
0 2 4 6
0 0.5 1
t = 0 t = 200 t = 800 t = 4000
Steady State
x P(x)
Figure 4.16: The simulation from t = 0 ∼ 4000 for the dynamical state of PDF and analytical solution with parameters: a = 2, ϵ = 2× 10−4, γ2 = 2× 10−3, and k = 1.
0 0.5 1 1.5 2 2.5 0
0.5 1 1.5
2 t = 0
t = 75 t = 500 t = 1800
Steady State
x P(x)
Figure 4.17: The simulation from t = 0 ∼ 1800 for the dynamical state of PDF and analytical solution with parameters: a = 2, ϵ = 2× 10−4, γ2 = 2× 10−3, and k = 10.
0 2 4 6 8 10 12
0 0.5 1
t = 0 t = 10 t = 50 t = 135 t = 1000
Steady State
x P(x)
Figure 4.18: The simulation from t = 0 ∼ 1000 for the dynamical state of PDF and analytical solution with parameters: a = 2, ϵ = 0.02, γ2 = 2× 10−3, and k = 1.
0.2 0.4 0.6 0.8 0
0.02 0.04 0.06 0.08
m P(m)
P
1(m) Numerics P
Figure 4.19: The probability distributions predicted by Eq. (4.65) and numerical results with the fitness function and parameters: N = 100, f (m) = em2, and γ = 1.
0.2 0.4 0.6 0.8
0 0.02 0.04 0.06 0.08
m P(m)
P
1(m) Numerics P
Figure 4.20: The probability distributions predicted by Eq. (4.65) and numerical results with the fitness function and parameters: N = 100, f (m) = e2m2, and γ = 2.
0.5 0.6 0.7 0.8 0.9 0
0.05 0.1
m P(m)
P
1(m) Numerics P
Figure 4.21: The probability distributions predicted by Eq. (4.65) and numerical results with the fitness function and parameters: N = 100, f (m) = e2m2, and γ = 1.
Table 4.1: The comparison of our results among P (m), P1(m), and numerics for the fitness function and parameters: f (m) = em2, γ = 1, N = 100.
m 0.44 0.46 0.48 0.50 0.52 0.54
P (m) 0.0672 0.0738 0.0782 0.0798 0.0782 0.0735 P1(m) 0.0691 0.0748 0.0780 0.0784 0.0757 0.0702 Numerics 0.0692 0.0749 0.0781 0.0785 0.0758 0.0703
Table 4.2: The comparison of our results among P (m), P1(m), and numerics for the fitness function and parameters: f (m) = e2m2, γ = 2, N = 100.
m 0.44 0.46 0.48 0.50 0.52 0.54
P (m) 0.0672 0.0738 0.0782 0.0798 0.0782 0.0735 P1(m) 0.0720 0.0764 0.0782 0.0771 0.0730 0.0663 Numerics 0.0721 0.0766 0.0783 0.0771 0.0730 0.0662
Table 4.3: The comparison of our results among P (m), P1(m), and numerics for the fitness function and parameters: f (m) = e2m2, γ = 1, N = 100.
m 0.68 0.70 0.72 0.74 0.76 0.78
P (m) 0.0722 0.0894 0.1036 0.1118 0.1117 0.1028 P1(m) 0.0744 0.0906 0.1033 0.1099 0.1086 0.0991 Numerics 0.0746 0.0908 0.1036 0.1103 0.1089 0.0993
Figure 4.22: The temperature distributions for pure Cu predicted by Eq. (4.69) for narrow region near the surface in (a) and for wider region in (b) with the parameters k0 = 401 W/m·K, q = 0.772×103 ∼ 2.28×103W/mm3, As = 800 mm2, L = 1 mm, l = 5 µm, Tb = 398 K, and Tt = 358 K. The temperature distributions for 304 stainless steel predicted by Eq. (4.69) for narrow region near the surface in (c) and for wider region in (d) with the parameters k = 14.9 W/m· K, q = 0.773 × 103 ∼ 2.28 × 103 W/mm3, As = 800 mm2, L = 1 mm, l = 5 µm, Tb = 378 K, and Tt = 318 K. The different colored lines correspond to various percentages of kinetic energy loss which is converted into the heat energy of sample.
Figure 4.23: The temperature distributions for pure Cu predicted by Eq. (4.71) for narrow region near the surface in (a) and for wider region in (b) with the parameters k0 = 401 W/m·K, q = 0.772×103 ∼ 2.28×103W/mm3, As = 800 mm2, L = 1 mm, l = 5 µm, Tb = 398 K, and Tt = 358 K. The temperature distributions for 304 stainless steel predicted by Eq. (4.71) for narrow region near the surface in (c) and for wider region in (d) with the parameters k = 14.9 W/m· K, q = 0.773 × 103 ∼ 2.28 × 103 W/mm3, As = 800 mm2, L = 1 mm, l = 5 µm, Tb = 378 K, and Tt = 318 K. The different colored lines corresponds to various percentages of kinetic energy loss which is converted into the heat energy of sample.
Figure 5.1: (a) The cross-sectional SEM micrograph taken from the sample subject to SMAT with the 2 mm flying balls and 40 µm SMAT amplitude. (b) The gradient variation trend of hardness of selected SMAT 304 SS samples.
Figure 5.2: The relationship between the resulting grain size and Zener-Holloman pa-rameter with the sample processed by different SMAT conditions.