The Excess Heat Capacity in Glass-forming Liquid Systems Containing Molecules Abstract

Transcription

1 The Excess Heat apacity in Glass-forming Liquid Systems ontaining Molecules H. B. Ke,. Wen *, W. H. Wg Institute of hysics, hinese Ademy of Sciences, Beijing ,. R. hina Abstract The excess heat pacity at glass trsition temperature in two types of glass-forming systems of [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 (0 x 1) d a(no 3 ) yh O (4 y 13) is studied. In the former system, with the replacement of K + tion with Na + tion, the excess heat pacity is almost invariable around 65.1 J mol -1 K -1, while the excess increases by 38.9 J mol -1 K -1 with the increasing per molar H O content in latter system. A qutitative description of the excess heat pacity is built up with the consideration of atomic d molecular trslational motion in the glass-forming systems. This finding might offer further understding to the glass trsition. AS: , ph, 65.0.Jk * ontact author, pwen@aphy.iphy.ac.cn 1

2 I. Introduction The excess of the specific heat of glass-forming liquids relative to rigid glass or crystalline materials are of particular importce to understd the mysterious glass trsition [1-8]. Due to the nonzero Δ existing widely in glass-forming liquids with few notable exceptions like fused quartz [1, ], a nonzero temperature T K lled as Kauzamn temperature, always exist to satisfy the equation, T m Δ S f = Δ dt, T T K where Δ S f d T m are the fusion entropy d the melting point [1]. The Kauzamn s paradox implies that the glass trsition is not only a kinetic process, but also underlying thermodynamic phase trsition. It is consistent with the empiril relation T K T 0 holding for my glass-forming liquids (GFLs) [9, 10]. T 0 is Vogel-Fulcher temperature, where the structural relaxation time or viscosity of GFLs is instable hypothetilly. orrespondingly, the Δ has always been described with the dynamic behaviors of GFLs in popular models [1, 11-15] to glass trsition. According to the entropy model [15], the Δ is given as ds onf / d lnt, where S onf is the configurational entropy. Unfortunately, in theory it is too difficult to separate the entropy into vibrational d configuratinal parts [16]. In the free volume theory [14], the Δ is related to the chge of the free volume with temperature. However, the free volume in principle is not measured qutitatively, d the qutitative description of the Δ based on free volume for GFLs is impossible. To describe the heat pacity of GFLs is still a problem unresolved. According to statistil physics, heat pacity of y matter in equilibrium in principle has a close relation with its intrinsic motions [17]. A basic characteristic of a liquid is well known [18]: A liquid n not support y shear within the normal experimental time sle. If atom or a molecule in the liquid state is considered at y moment, it will be interacting with the neighbors, vibrating as though it were atom or a molecule in the solid state. At the next moment with certain gap in time to the previous moment, the atom/molecule may already move away from its previous site. Obviously, the processed involve atomic/molecular trslational motion, d

3 need overcome energy barrier ΔE, d are activated thermally. The average ΔE in general at constt pressure depends on temperature, d the structural relaxation time in a GFL following the general Arrehnius form n obey the Vogel-Fulcher equation ΔE D like: τ = τ = 0 exp τ 0 exp, where k B is Boltzmn s constt d D is a kbt T T0 constt. It is consistent with the current elastic model [19] where the barrier for a flow event must be considerably larger th the thermal energy as temperature is approaching to the glass trsition temperature. Then, a glass is formed as τ is so large that the trslational motions n not be observed within the experimental time sle. In other words, within the experimental time sle the trslational motions are available in GFLs, but the corresponding glasses. Accordingly, the Δ of glass-forming liquids might be related to the trslational motions. In this letter, we mage to build up a correlation between the excess motions in GFLs relative to their glassy state d the Δ at T g in two typil [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 (0 x 1) d a(no 3 ) yh O (4 y 13) GFL systems. Our experimental results confirm the relation. II. Experimental We chose [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 (0 x 1) d a(no 3 ) yh O (4 y 13) glass forming systems since they offer a good opportunity to understd the origin of the Δ for the systems containing atoms d simple molecules. Besides the good glass-forming ability d a clear Δ at T g, the motions of the compositional units n be well clarified. The materials were formed directly by the melting of mixtures of deionized water, NaNO 3, KNO 3, d a(no 3 ) 4H O. Reagent grade NaNO 3, KNO 3, d a(no 3 ) 4H O were obtained from Sigma-Aldrich, which were directly used without further purifition. The detail preparation of 6[xNaNO 3 (1-x)KNO 3 ] 4[a(NO 3 ) ] materials d samples used in the measurements of differential snning lorimeter (DS) has been reported in Ref.[0]. The a(no 3 ) yh O materials with the mass of 5 g were sealed into glasses vials. In order to homogenize the mixture, the vials were hold at 333 K for 3

4 15 min. Thermal alysis occurring during both cooling d heating was rried out using a Mettler Toledo DS1 thermal alyzer. The mass of the samples used for DS measurements are around 15 mg. III. Results d Discussion Figure 1 shows the DS curves for the [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 (0 x 1) materials. All of curves were measured at a heating rate of 0K/min. The measurements followed a cooling from the temperature (T g +0 K) at 0 K/min for each DS sample. The system n be regarded as a binary mixture. otassium, sodium d lcium ions have spheril charge distributions d different from nitrate ions of trigonal shape [1]. It has been reported that the glass-forming ability of [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 (0 x 1) systems become worse with the increase of x [0]. Even so, a clear glass trsition with obvious Δ d supercooled liquid region still exists for all of compositions. The T g, determined by the usual way (see Fig.1), is found to depend on the composition. With the replacement of the first alkali ionic (K + ) by the second alkali ionic Na +, T g for the system decreases gradually to a minimum as x approaching to 0.5. A similar phenomenon exists in other system [] d represents the alkali-mixed effect [3]. The fragility index m of GFLs n be given by DS measurements [4]. The determined values of T g, Δ at T g d m for the system are listed in Table I. The x dependence of the fragility index m d the Δ of the [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 system are displayed in Fig.. For [KNO 3 ] 60 [a(no 3 ) ] 40, KN, with extreme x = 0, the value of m is equal to 107. The value of 107 is higher th the value m = 93 reported by Bohmer et al.,[5], d less th the value 15 obtained from the temperature dependence of viscosity close to T g [6]. The value of m for the system depends on the composition, d the tendency of m with x is similar to that of T g. As x tends to 0.5, m approaches to a minimum. In contrast, the Δ p is almost independent of the composition d close to a constt value of 65. J K -1 mol -1. This implies that in this system no relation between the Δ p d the m exists. Therefore, the conventional wisdom that GFLs 4

5 with large m has large Δ p at T g [1, 7], n not be applied in this GFL system. A similar result has been found in oxide glass-forming systems []. As a binary system, the rriers of the motions only have two types. One type contains K +, Na +, d a + ions. Another involves NO - 3 ions. The contributions of K +, Na +, d a + ions to the heat pacity are equivalent since the Δ p is invariable as K + is replaced by Na + in system. It is appropriate to separate the Δ p into tionic d ionic parts. Defining the related to one mole tions d the for one mole ionic NO 3 -, the Δ p in [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 system n be expressed as: Δ = = 64.9JK 1 mol 1. To determine the values of the d, other equation containing the d must be provided. Figure 3 shows the DS curves for a series of a(no 3 ) yh O (4 y 13) samples. The DS measurements are rried out at a heating of 10 K/min. The values of T g d the Δ p at T g for all of materials are listed in Table I. In this system, T g decreases obviously with the increasing H O content, but the Δ p increases. The values of T g d Δ p for a(no 3 ) 4H O are 1 K d 34.3 J K -1 mol -1. We note that the values are a little smaller th those reported previously [7]. It is due to the different measurements used with different samples. Importtly, we will show the compositions, especially the H O content, have a great effect on the Δ p (see Table I d Fig.4). The large value of Δ p has been explained since its m (around 100) is large d its GFL is a typil fragile one [7, 8]. The value of m, compared with that of KN, is still smaller, but a(no 3 ) 4H O has much larger Δ p th KN has. It n not be explained well by the conventional wisdom [1, 7]: the more m is, the larger Δ p is. Different from the conventional wisdom, we think the key reason is that the particles involving a +, NO - 3, d H O in one molar a(no 3 ) 4H O sample is more th those in one molar KN. Following the description of the Δ p in [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 system, the Δ p for a(no 3 ) yh O system is written as HO + y +, where H O is related to one molar H O molecules. In a(no 3 ) yh O systems, the 5

6 only variable is H O content. The chge Δ p in the system must be proportional to the chge of y. Moreover, the values of the H O d the sum + n be deduced from the linear relation between the Δ p d y. The linear increase of the Δ p with the increase of y in the system is shown in Fig. 4. The slop of the linear relation is 37.9±1.1 J mol -1 K -1, d its intercept on vertil axis is 87.6±10 J mol -1 K -1. So, the value of H O is 37.9 J mol -1 K -1, d the sum of + is equal to 87.6 J mol -1 K -1. ombined with the result, = 64.9JK mol, the values of the d are determined to be 1.0 d 37.8 J mol -1 K -1. Remarkably, it is found that the value of the, 3 R (around 1.5 J mol -1 K -1 ), error, respectively. 9 R d (around 37.4 J mol -1 K -1 ) d H O n be written as 9 R with a small The following question is what contributes to the, d H O. It is usual, in theoretil considerations, to ignore the difference between the at constt pressure d V at constt volume for a condensed matter; this neglect involves only small errors, d n be remedied [9]. In theory, the V n be lculated directly from the motions of particles, whose additive integrals completely define the statistil properties of a closed system [17]. The motions of particles in a liquid involve vibrations, rotations d trslations. learly, atomic vibrations in the GFLs have nothing with the. The vibrations for one atom have six degrees of 6 freedom, d contribute kb to the heat pacity, d the atomic vibration n exist in both of glass d liquid. The remaining atomic motions, contributing to the 6, are atomic trslational motions. These motions are related to the atomic jumps away from their positions in the supercooled liquid state. In space, each atomic trslational motion has three independent directions with the same possibility. So, this motion has three degrees of freedom. The corresponding heat pacity arisen from atomic trslation is 3 R, d exactly equal to the experimental value of.

7 It is appropriate to think that the trslational contribution to the heat pacity is not measured in glass region since the structural relaxation of equilibrium liquid n not exist in the normal DS experimental time. Therefore, is determined by the atomic trslation. ompared to atomic motions, the motions of atomic group d H O molecule are a little complex, but still clear. has not only vibration, but also rotation d trslation. The vibration related to atomic group involves the vibration of the atomic group d atoms inside the atomic group. The former has six degrees of freedom, d the latter has twelve degrees of freedom. So, the 9 ( R ) does not correspond to the vibration. The rotation of, involving the rotation of trigonal plate d the rolling of oxygen around nitrogen inside the atomic group, has three degrees of freedom in totally, d n not contribute to either. The trslation of atomic group has nine degrees of freedom, but three. It is due to that one atomic group has three molecular forms to trslate in space. The three molecular forms correspond to its three degrees of vibrational freedom. One atomic group with a given molecular form has three degrees of trslational freedom. So, the trslation of one molar atomic group contributes 9 R to the heat pacity, d is the origin of the atomic group, the correlation between. Same as H O d the trslation of H O molecule also exists. One H O molecule, regarded as a rigid bent, has three degrees of rotational freedom. Its trslation in liquid has nine degrees of freedom, d contributes 9 R to the heat pacity. The effect of molecular rotation on its form to trslate in liquid state is due to that the particles have strong interactions with their neighbors. rior to trslation, rotation already takes place. It is known that the 7

8 activation of molecular rotations is easier th that of the trslations in condensed state [30, 31]. Our finding offers a clear picture to the glass trsition in GFLs with obvious Δ. Upon cooling (see Fig. 3), across the glass trsition the rapid decrease in heat pacity is accompied by the gradual disappearce of trslations in the GFLs within the DS experimental time sle. So, the glass trsition is a pure kinetic process. The process corresponds to the disappearce of trslations within the experimental time sle. Since T g depends on the experimental time sle, in theory T g n be y temperature in liquid region below T m. The contribution of the trslation to heat pacity at constt pressure is independent of temperature. Thus, the value of the Δ n be predicted to be invariable with temperature. The finding gives also strong evidence to idea that the excess entropy of GFLs is only related to the trslational motions. It is not necessary to assume that the excess entropy has a close relation with the configurational entropy. Even if the experimental time sle is long enough d T g should tend to lower temperature, no thermodynamic phase trsition could take place at lower T g. The frozen of the trslation across the glass trsition is just a dynamic process on a certain experimental time. IV. onclusion In summary, the clear correlation of the excess heat pacity with the corresponding trslational motion in [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 (0 x 1) d a(no 3 ) yh O (4 y 13) glass-forming systems is built up. The results show that within the DS experimental time sle all of trslational motions are available in supercooled liquids, but in glasses. The glass trsition in the two glass-forming systems is related to the frozen of trslational motions of all rriers in the GFLs. The work was supported by the Science Foundation of hina (Grt Nrs: d ) d MOST 973 of hina (No. 007B613904, 010B731603). 8

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11 aptions Figure 1. The DS curves of [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 ( 0 x 1) glassy samples measured during a heating at 0 K/min. Figure. The x dependence the Δ p at T g d the fragility index m for [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 glass-forming system. Figure 3. The DS curves for series of a(no 3 ) yh O ( 4 y 13) glassy samples measured during a heating at 10K/min. The dash lines are the curves measured during a cooling at 10K/min. Figure 4. The linear relation between the Δ p at T g d y in a(no 3 ) yh O system. The slope of the fitting line is 37.8 J mol -1 K -1, d interact on vertil axis is 87.6 J mol -1 K -1. Table I. The compositions, glass trsition temperature T g, the fragility index m d the jumps in heat pacity Δ p at T g for [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 (0 x 1) d a(no 3 ) yh O (y = 4~13) glass-forming systems. 11

12 Table I. The values of T g, Δ at T g d m determined by DS for the a(no 3 ) yh O d [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 ( 0 x 1) materials. a(no 3 ) yh O [xnano 3 (1-x)KNO 3 ] 60 [a(no 3 ) ] 40 y T g (K) Δ p x T g (K) m Δ p (J mol -1 K -1 ) (J mol -1 K -1 )

13 Heat flow ( a.u. ) Endoth. x : 1 ; 0.875; 0.75 ; 0.65; 0.5 ; 0.375; 0.5 ; 0.15; 0. T g Temperature ( K ) Figure 1. Ke, et. al. 13

14 10 80 m Δ ( JK -1 mol -1 ) x 50 Figure. Ke, et. al. 14

15 y : Heat flow ( a.u. ) Endoth Temperature ( K ) Figure 3. Ke, et. al. 15

16 600 Δ ( JK -1 mol -1 ) y Figure 4. Ke, et. al. 16