Hall–Petch relationship in nanometer size range

Journal of Alloys and Compounds 361 (2003) 160–164

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Hall–Petch relationship in nanometer size range M. Zhao, J.C. Li, Q. Jiang* Key Laboratory of Automobile Materials of Ministry of Education and Department of Materials Science and Engineering, Jilin University, Changchun 130025, PR China Received 23 January 2003; received in revised form 25 March 2003; accepted 25 March 2003

Abstract The effect of melting temperature on the Hall–Petch relationship has been studied. As grain size decreases, the melting temperature of the nano-structured crystals also decreases, therefore the Hall–Petch relationship has its limitation and is no longer sufficient when the grain size decreases to about the 15–30 nm range. When the yield strength or hardness is taken as a function of the reciprocal of the square root of the grain size, it has a numerical maximum whose location depends on the size of the bulk melting enthalpy of the crystals. Experimental results agree well with the modification induced by the size-dependence. In addition, the appropriate size range for the classic Hall–Petch relationship is discussed in terms of the modification of the relationship.  2003 Elsevier B.V. All rights reserved. Keywords: Metals; Nanostructured materials; Mechanical properties

1. Introduction It is well known that as the grain size of crystals D decreases, the yield strength s or hardness H of the crystals increases. This relationship is the empirical Hall– Petch relationship [1,2]:

s (D) 5 s 09 1 k d9 D 2n

(1)

where n has a typical value of 0.5. On some other occasions, values close to 1 have also been suggested [3]. For normal crystalline materials with D .10 mm where the D 21 / 2 term in Eq. (1) is negligible, s 09 5 s (D → `). k d9 , a positive material constant, depends on the resistance of grain boundaries to dislocation movement. When 10 mm. D .100 nm, the model prediction of Eq. (1) is in reasonable agreement with experimental results. However, when D ,100 nm, such as 15–30 nm, depending on the nature of the materials, the prediction of Eq. (1) is no longer appropriate to predict experimental results [3–13]. Eq. (1) implies that the Hall–Petch relationship is temperature-independent. This is mostly due to: (1) the normal experimental temperature T d is room temperature [3–14]; (2) for Hall and Petch investigations, 100 nm is almost the smallest grain size of bulk materials. In that *Corresponding author. Fax: 186-431-570-5876. E-mail address: [email protected] (Q. Jiang). 0925-8388 / 03 / $ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016 / S0925-8388(03)00415-8

range, the melting temperature of crystals, T m (D), is not size-dependent [15–17]. Both assumptions lead T d /T m (D) to be a constant. When D decreases to less than 100 nm, T m (D) evidently drops [15–17]. Therefore, T d /T m (D) increases even when T d remains constant and the crystals start to have high temperature properties where the contribution of grain boundary movement on plastic deformation is present [3]. As a result, s 09 and k 9d become temperaturedependent or size-dependent. As D further decreases, a zero or even a negative k d9 has been reported [18–20]. To adequately predict the size-dependent yield strength, the effects of grain size or melting temperature of crystals on coefficients s 90 and k 9d must be considered. When D is a constant, let s 099 denote the effect of T d on yield strength, we have [21]:

s 99 0 5 k9 t exp f Q /s2RT ddg

(2)

where k 9t is a constant, Q is the activation energy for interface migration, R is the ideal gas constant. According to Ref. [22]: Q~RT m (D)

(3)

where T m (D) has been used to substitute the original T m0 denoting the bulk melting temperature [22]. As a first order approximation, it is assumed that s 09 5 s0 1 s 099 , or s 90 is a sum of the temperature-independent term and the temperature-dependent term. Accordingly, substituting Eq. (3)

M. Zhao et al. / Journal of Alloys and Compounds 361 (2003) 160–164

into Eq. (2) and combining the coefficient of Eq. (3) into k t9 of Eq. (2), it reads:

S

T m (D) s 90 5 s0 1 k 9t exp ]] 2T d

D

(4)

T m (D) in Eq. (4) has been determined [15–17], which is expressed as:

F

T m (D) 2 2Svib /(3R) ]] 5 exp ]]]] T m0 D/D0 2 1

G

2. Model It is assumed that k 9d in Eq. (1) has the same temperature dependence as in Eq. (4): k 9d 5 k 99 d exp fsT m (D) / 2T ddg

(6)

This is because the migration resistance of dislocations should have also an Arrhenius relationship with temperature. Plug Eqs. (4) and (6) into Eq. (1) with n51 / 2: 21 / 2 s (D) 5 s0 1sk 9t 1 k 99 d d D

F

F

measurements of the materials, Eq. (7) can be further simplified. Since e 2x ¯ 1 2 x, when x is small, the double 2 2S m / (3R ) exponential term in Eq. (7), exp f ]]] , is approxiD / ( 6h) 2 1 g 2S m / (3R ) mately equal to 12 ]]] . Accordingly, Eq. (7) is D / (6h) 2 1 rewritten as:

F

1 Hm /(3R) s (D) 5 s0 1 f k t 1 k d D 21 / 2 g exp 2 ] ]]]] T d D/(6h) 2 1

G (8)

(5)

where Svib is the vibrational part of overall melting entropy Sm . Except semiconductors, Svib ¯Sm due to small contributions on Sm , such as the configurational part and electronic part of melting entropy [17]. For a solid solution, h is the mean diameter of the atoms in the solution. D0 denotes a critical diameter where almost all atoms of a grain are located on its interface. For nanostructured materials, D0 56h [15]. For elements and compounds, h denotes atomic or molecular diameter (if lattice constants in different directions of the molecules are different, h is the mean value as a first order approximation). To obtain more reliable yield strength under both compression and tension, further experiments focused on removing flaws (voids, microcracks, incomplete boundaries of particles, etc.). Such presence of flaws increase as grain size D drops and they lead to decrease of s values that are lower than the corresponding intrinsic values [4,18,19,23]. Thus, in the comparison between the experimental results and numerical models, the newer experimental results are taken into account to avoid these effects on s. In this contribution, Eqs. (4) and (5) are substituted into Eq. (1) to consider the melting temperature effect on both coefficients of s 09 and k d9 . The results show that the modified s (D) functions for metallic elements of Cu, Fe, Ni, Pd and Zn, alloy Ni–P, compounds NiZr 2 and TiO 2 correspond to experimental results well.

Tm 0 2 2Sm /(3R) 3 exp ]] exp ]]]] 2T d D/(6h) 2 1

161

GG

(7)

When D .10 nm, which is often the case for s (D)

where k t 5 k 9t exp[T m0 /(2T d )], k d 5 k d99 exp[T m0 /(2T d )], Hm 5 T m0 Sm is the bulk melting enthalpy. As D →`, it leads to:

s 90 5 s0 1 k t

(9)

and k d9 5 k d

(10)

To determine the relative size of s0 and k t , let ds (D) / d(D 21 / 2 ) 5 0 in Eq. (8), where s 5 smax with the subscript max denoting the maximum: k T d R 1 / 2 3T d R 1 Hm 21 / 2 9hT d R 23 / 2 ]t 5 ]] D 2 ]]]] D max 1 ]] D max k d 4hHm max Hm Hm (11) /2 Since most values of D 21 max in the experimental results are located among 0.2–0.4, while h values are about 0.3 nm, Hm /(T d R)¯4–7 (let T d 5300 K):

k t /k d ¯ 0–0.1

(12)

With Eqs. (9), (10) and (12), the fitting parameters among s0 , k d and k t in Eq. (8) are comparable with the original fitting parameters of s 90 and k 9d in Eq. (1). This comparison can be used to verify the fitting parameters of Eq. (8).

3. Results Figs. 1–5 show experimental data and s (D) functions in terms of Eq. (8) for elements Cu, Fe, Ni, Pd and Zn, Fig. 6 for Ni–P alloy, and Figs. 7 and 8 for compounds NiZr 2 and TiO 2 . When there are fitted values of s 90 the in experimental measurements, these s 90 values are directly taken in our fitting, which is the case of Figs. 1–4. However, our estimated k d values are more or less different from the corresponding literatures due to different function forms between Eq. (1) and Eq. (8) as shown in the corresponding figure captions. To clarify, these fitted parameters from the corresponding literatures in terms of Eq. (1) are also shown in the figures. Although our fitting results with the above phenomenological model are only roughly in agreement with the experimental results, the tendency that there is a maximum in the s (D) function in the range of 15–30 nm implies that the interplay between dislocation and grain-boundary processes occurs during

M. Zhao et al. / Journal of Alloys and Compounds 361 (2003) 160–164

162

Fig. 1. Comparison of Hall–Petch relationship of Cu between the modeling (solid line) of Eq. (8) and experimental results shown as j [26] and s [13]. In our fitting, s 09 519.1 MPa is the experimental result [26], s0 , k t and k d are obtained by fitting all experimental results with the least-square method in terms of Eq. (8), where s0 1 k t 5 s 09 in terms of Eq. (9). During the fitting, necessary parameters in Eq. (8), see Table 1 with T d 5300 K. As result, s0 59.5 MPa, k t 59.62 MPa and k d 54326 21 / 2 21 / 2 MPa nm . Note that k d in experiments [26] is 5439 MPa nm , which differs from our fitted k d value of about 20%. The dash line shows Eq. (1) with the experimental parameters of s 90 and k d [26].

deformation and their contributions on s vary as size changes [33–36]. Namely, as D decreases to the size order of 10 nm, grain-boundary weakening is present [36] and the s (D) value drops gradually. For large D (bulk limit), Eq. (8) can be further approximated by:

Table 1 h and Hm values of elements, metallic alloys and intermetallic compounds used in Eq. (8) Materials

T m0 (K)

h (nm)

Hm (kJ g-atom 21 )

Cu Fe Ni Pd Zn Ni 80 P20 NiZr 2 TiO 2

1358 1809 1726 1825 693 1184 1393 2098

0.2826 0.2822 0.2754 0.3040 0.3076 0.2429 0.4681 0.3860

13.05 13.80 17.47 17.60 7.322 15.34 14.47 15.90

h and Hm of elements are obtained from Refs. [24,25]. h of Ni 80 P20 alloy is defined as h5(0.8h Ni 10.2h p ), where h Ni 50.2492 nm [24] and h p 50.2180 nm [24]. T 0 and Hm of Ni 80 P20 alloy are obtained from Ref. [32]. NiZr 2 has a body-centered tetragonal C16(Al 2 Cu) structure with ] a50.6499 nm and c50.5241 nm. h 5Œ2 / 2[(2a 1 c) / 3]. TiO 2 has a simple tetragonal C4 lattice with a50.4593 nm and c50.2959 nm. h 5 (2a 1 c) / 3. Hm values of NiZr 2 and TiO 2 are cited from Ref. [30] and Ref. [31], respectively.

Fig. 2. Comparison of Hall–Petch relationship of Fe between the modeling (solid line) of Eq. (8) and experimental results shown as h [26], m [26], d [4] (the original data of hardness is transformed to yield stress through the relationship of s 5 Hv / 3). s 90 5103 MPa and k d 5 21 / 2 15178 MPa nm [26]. The detailed fitting process sees the caption of Fig. 1. The necessary parameters in Eq. (8), see Table 1 with T d 5300 K. The obtained fitting parameters are s0 575 MPa, k t 528.5 MPa and k d 512437 MPa nm 21 / 2 . Note that k d in experiments [26] is about 18% different from our fitted k d value. The dash line shows Eq. (1) with the experimental parameters of s 90 and k d [26].

s (D) ¯ s0 1 k t 1 k d D 21 / 2 2 f 2hHm /sRT ddg f k t D

21

1 kdD

23 / 2

g

(13)

In terms of Eq. (13), k d appears both in D 21 term and D 23 / 2 term. The latter term should denote the temperature effect. Since k d .100k t , k t as a coefficient of D 21 has little effect on s (D). Note also that s0 . 10k t (see the captions of the figures), a weak effect of k t on s 90 is also evident. Thus, the change of s (D) 2 D 21 / 2 curve in Eq. (1) is essentially induced by temperature effect on k 9d . This analysis leads to a further simplifying of Eq. (13):

s (D) ¯ s 90 1 k 9d D 21 / 2 (1 2 c /D)

(14)

where c 5 2hHm /(RT d ), s 90 5 s0 1 k t , and k 9d 5 k d . Comparing Eqs. (1) and (14), it is obvious that Eq. (1) has neglected the term of cD 23 / 2 . Since c,10 nm, Eq. (1) has little error when D .100 nm. Eq. (14) implies that the maximum of the s (D) curve should be absent at D .100 nm. It is interesting to note that the Hm and T d are two parameters affecting the size-dependence of the yield stress as shown in Eq. (8). Since T d is typically the room temperature, Eq. (8) suggests that a material with a higher /2 Hm value has a larger smax , yet a smaller D 21 max . It is understood that a material with a higher Hm value has a

M. Zhao et al. / Journal of Alloys and Compounds 361 (2003) 160–164

Fig. 3. Comparison of Hall–Petch relationship of Ni between the modeling (solid line) of Eq. (8) and experimental results shown as m [8], s [18], \ [18] and j [27]. s 90 51092 MPa and k d 526281 MPa nm 21 / 2 [8]. The detailed fitting process sees the caption of Fig. 1. The necessary parameters in Eq. (8), see Table 1 with T d 5300 K. The obtained fitting parameters are s0 51021 MPa, k t 571 MPa and k d 526632 MPa nm 21 / 2 . Note that k d in experiments [8] is about 1% different from our fitted k d value. The dashed line shows Eq. (1) with the experimental parameters of s 90 and k d [8].

stronger grain boundary strengthening as D decreases, which in return leads to an early grain boundary softening at larger D value.

Fig. 4. Comparison of Hall–Petch relationship of Pd between the modeling (solid line) of Eq. (8) and experimental results shown as j [9] and m [28]. s 09 51023 MPa [9]. The necessary parameters in Eq. (8), see Table 1 with T d 5300 K. The obtained fitting parameters in terms of Eqs. (8) and (9) are s0 5939 MPa, k t 584 MPa and k d 512983 MPa nm 21 / 2 .

163

Fig. 5. Comparison of Hall–Petch relationship of Zn between the modeling (solid line) of Eq. (8) and experimental results shown as j [29]. The fitted parameters from the experimental data in terms of Eqs. (8) and (9) are s0 5120 MPa, k t 519 MPa and k d 53015 MPa nm 21 / 2 . The necessary parameters in Eq. (8), see Table 1 with T d 5300 K.

4. Conclusions In summary, a modified Hall–Petch relationship is established based on a model for the interface migration of atoms and a model for the size-dependent melting temperature of crystals. In light of the modification, the abnormal Hall–Petch relationship is modified by the increasing of the ratio of T d /T m (D) as D decreases. Therefore, the nanostructured crystals gradually start to have high tem-

Fig. 6. Comparison of Hall–Petch relationship of Ni 80 P20 alloy between the modeling (solid line) of Eq. (8) and experimental results shown as d [5]. The fitted parameters from the experimental data in terms of Eq. (8) are s0 53940 MPa, k t 558 MPa and k d 512303 MPa nm 21 / 2 . The necessary parameters in Eq. (8), see Table 1 with T d 5300 K.

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Acknowledgements The financial support of the NNSFC under Grant No. 50025101 and 50071023 are acknowledged.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] Fig. 7. Comparison of Hall–Petch relationship of NiZr 2 between the modeling (solid line) of Eq. (8) and experimental results shown as j [11]. The fitted parameters from the experimental data in terms of Eq. (8) are s0 51024 MPa, k t 561 MPa and k d 535267 MPa nm 21 / 2 . The necessary parameters in Eq. (8), see Table 1 with T d 5300 K.

perature properties. The modeling of the modified Hall– Petch relationship agrees well with the experiment data of elements, alloys and compounds.

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

Fig. 8. Comparison of Hall–Petch relationship of TiO 2 between the modeling (solid line) of Eq. (8) and experimental results shown as j [6]. The fitted parameters from the experimental data in terms of Eq. (8) are s0 53780 MPa, k t 5132 MPa and k d 521816 MPa nm 21 / 2 . The necessary parameters in Eq. (8), see Table 1 with T d 5300 K.

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