Low-Temperature Excess Heat Capacity of Potassium Germanate Glasses

Low-temperature heat capacity of potassium germanate glasses (xK2O·(100-x)GeO2; x=0.0, 10.1, 19.0, 28.2, 39.0) (x indicates K2O mol% content) has been measured in the temperature range from 2 to 50 K with K2O content. From a result of the heat capacity Cp, it has been found that an excess heat capacity is not caused by a regular thermal motion but an interaction. In addition, it has also been found that a relationship between a maximum of reduced excess heat capacity CpT max and elastic modulus is dual. Moreover, a ‘hole’ model of liquid theory was applied to explain the formation of resonant mechanism. This model leads us to an idea that the excess heat capacity is described by degree of freedom of reallocated-and-isolated-structural units. Consequently, we conclude that the excess heat capacity is caused by the reallocated-main-network structure containing holes resonant with the reallocated-and-isolated-microstructural units.


Purpose of Reseach
The purpose of reseach is to measure the heat capacities C p of potassium germanate glasses in the range of temperature from 2 to 50 K with K 2 O content and to clarify an origin of the excess heat capacity compared with the Debye model from a viewpoint of the microstructure.

Historical Background
We follow the route of the research on low-temperature thermal properties for the past fifty years or so.First, in 1971, Zeller andPohl (1971) have revealed universal properties of amorphous solids including glasses in low-temperature.They measured the specific heat and thermal conductivity of vitreous silicate-and germanate-system and selenium in the temperature range from 0.05 to 100 K.They found that the low-temperature thermal properties of amorphous solid are different from those of crystalline solid.It was found that the thermal conductivity varies as T 1.8 below T=1 K and the specific heat varies as AT+BT 3 between 0.1 and 1 K (A and B are constants).In addition, the excess specific heat and the plateau of thermal conductivity around 10 K were also discussed.The following year 1972, Anderson, Halperin and Varma (1972) and Phillips (1972) proposed a tunneling model respectively, which explains the anomaly of the specific heat and thermal conductivity in amorphous solid below 1 K. From 1980 to 2000, the excess heat capacity and plateau of thermal conductivity at about 10 K were explained using the extension of the tunneling model in combination with Rayleigh scattering, sound waves and soft localized vibrations by Yu and Freeman (1987), Buchenau, Galperin, Gurevich, Parshin, Ramos and Schober (1992) and Gil, Ramos, Bringer and Buchenau (1993) respectively.Since 2000, the excess heat capacity at about T=10 K has been discussed separately from the thermal property below 1 K.For example, in 2003, the heat capacity of B 2 O 3 and GeO 2 glasses were measured from less than 10 to 350 K and at 0 K their excess entropies were calculated by Richet, Ligny and Westrum (2003).In addition, the relationship between the importance of (calorimetric) Boson peak and glass transition temperature was mentioned.In 2009, the vibrational density of states was represented from inversion of low-temperature heat capacities of vitreous SiO 2 and a series of Li, Na and K silicate glasses from 10 to 300 K by Richet (2009).The relationship between the vibrational density of states and the heat capacity of glasses was discussed.In 2010, the effects of the substitution of nitrogen for oxygen on the heat capacity and vibrational entropy of several yttrium aluminosilicate glasses with Si 3 N 4 contents have been investigated by Richet, Rouxel, Kawaji and Nicolas (2010).Of course, the vibrational density of states was also calculated and compared with heat capacity.Moreover, the relationship between (calorimetric) Boson peak and elastic modulus was analyzed.As mentioned above, a lot of papers have been dedicated to a problem about the energy in glasses but few papers have been dedicated to a problem about the microsturucture in glasses.Therefore, there is room for an intensive study of the problem about microstructure and we may acquire some useful information on the excess heat capacity.

Method
We study the relationship between two germanate anomalies on the basis of the microstructure.The microstructure of potassium germanate glasses has been obtained from our Raman scattering measurement (Mamiya, Matsuda, Fukawa, Kawashima, & Kojima, 2009).Two germanate anomalies mean they have a same K 2 O content at which each anomaly shows a maximum or a minimum of its physical property.We perform an experiment according to the following procedure.

1) Sample preparation
2) Low-temperature heat capacity measurement 3) To find the germanate anomaly about the excess heat capacity 4) To find the counterpart of the germanate anomaly about the excess heat capacity 5) To discuss and investigate the relationship between the two germanate anomalies on the basis of the microstructure 6) To clarify the origin of the excess heat capacity in the potassium germanate glass

Sample Preparation
Four glass samples were prepared in the series of xK 2 O•(100-x)GeO 2 ; x=10.1, 19.0, 28.2, 39.0.All the glasses were synthesized by the solution method (Kodama, Matsushita, & Kojima, 1995;Matsuda, Fukawa, Ike, Kodama, & Kojima, 2008).The advantage of this method is that the starting materials are initially made to react in an aqueous solution to achieve homogeneity.Analytical reagent-grade KOH•H 2 O and GeO 2 were used as the starting materials without further purification.The starting materials were made to react by adding distilled pure water in a Teflon beaker.The beaker containing the solution was then placed in a drying oven at 140 °C for 7 days.After the complete evaporation of water, a chemically synthesized powder was obtained.This powder was melted in a Pt crucible for 1.5 hours at about 950 to 1100 °C depending on the content.The homogenized bubble-free melts were cast in a graphite mold for bulk glasses and splat-quenched.For x=0.0, the germania (GeO 2 ) glass sample was prepared by M. Kodama.The method of GeO 2 glass preparation has been described by Zeller and Pohl (1971).The content of each glass was chemically analyzed (Kodama, Iizuka, Miyashita, Nagai, Clarida, Feller, & Affatigato, 2003).The analyzed value was used in this study.

Sample Size
The five splat-quenched glasses were ground to shape a block of about 2×2×1.5 (thickness) mm 3 for the low-temperature heat capacity measurement and preserved in a desiccator to keep free from moisture.

Low-Temperature Heat Capacity Measurement
The heat capacity of potassium germanate glasses was measured in the range of temperature from 2 to 50 K with K 2 O content using Physical Property Measurement System (PPMS) of Quantum Design ○ c at the Cryonics Div., Research Facility Center of Tsukuba University.The heat capacity was measured by the thermal relaxation method and calculated by subtracting an addenda measurement from the total capacity.The addenda measurement means the measurement of the heat capacity of the grease and the platform (Richet et al., 2010).

Results
Figure 1 shows the heat capacity C p of the potassium germanate glasses with K 2 O content between 2 and 50 K.The heat capacities were converted from a mol to a g atom bases to make consistent comparisons.The C p of germania (GeO 2 ) glass is good agreement with that of GeO 2 glass measured by P. Richet et al. (2003).

Reduced Excess Heat Capacity C p T -3
Figure 2 shows a reduced excess heat capacity C p T -3 compared with Debye T 3 laws.The C p T -3 of germania (GeO 2 ) glass is also good agreement with that of GeO 2 glass measured by P. Richet et al. (2003).Their C p T -3 graph of GeO 2 glass is described using mol bases.In Figure 2, every C p T -3 exhibits a maximum around 10 K.The C p T -3 of all glasses increases with increasing K 2 O content around T=50 K.However, the C p T -3 of GeO 2 glass increases and crosses those of the other three K 2 O content glasses with decreasing temperature T. This behavior is very similar to the silica (SiO 2 ) glass in sodium silicate system (Richet, 2009).max shows a minimum for x=15, this is the germanate anomaly about the excess heat capacity.

Counterpart of Germanate Anomaly about Maximum of Reduced Excess Heat Capacity C p T -3 max
We determine the counterpart of the germanate anomaly about the maximum of reduced excess heat capacity C p T -3 max .Figure 4 shows the elastic moduli we measured (Mamiya, Matsuda, Kaneda, Kawashima, & Kojima, 2010).The elastic modulus exhibits a maximum for x=15.Therefore, the elastic modulus is the counterpart of the germanate anomaly about the maximum of reduced excess heat capacity.Compared with Figure 3, the behavior of the elastic modulus is opposite to that of the maximum of reduced excess heat capacity C p T -3 max .We will mention about the resonance and a 'hole' model of liquid theory as the feedback from the experimental results in the next section.

Resonance
In Figure 2, every excess heat capacity C p T -3 reaches the peak around 10 K.This phenomenon seems to be a resonance (Halliday, Resnick, & Walker, 2001).Especially, the behavior of the GeO 2 glass is interesting.This means the GeO 2 glass has the microstructure which makes resonant intensity larger than the other three K 2 O content glasses.And there are two indispensable factors in the resonance.They are a forced (driven) oscillator and a free oscillator.The forced oscillator is considered as the regular thermal motion.Because Chumakov et al. (2011) have concluded that the density of states (DOS) shows that the glass and the relevant crystal have the

Relationship between Two Properties on the Basis of Microstructure
We discuss the relationship between the elastic modulus and the maximum of reduced excess heat capacity C p T -3 max on the basis of the microstructure from our Raman scattering measurement (Figure 5).For 0 ≤ x ≤ 20, the elastic modulus is described by superposition of two factors.The one factor is the number of 3-membered rings of GeO 4 tetrahedra.Because 6-membered ring of GeO 4 tetrahedra (for x = 0.0, Raman band at 420 cm -1 ) is broken down to two pieces of 3-membered rings of GeO 4 tetrahedra by K 2 O and the structure of 3-membered ring of GeO 4 tetrahedra is denser and harder than that of 6-membered ring of GeO 4 tetrahedra.The other factor is the number of GeO 6 octahedra.Because m-membered chain of GeO 4 tetrahedra containing Q 3 or Q 2 is changed into m-membered chain of GeO 4 and GeO 6 octahedra by K 2 O, the structure of GeO 6 octahedron is harder and more stable than that of Q 3 or Q 2 .There are a few m-membered chains of GeO 4 tetrahedra containing Q 3 or Q 2 in this K 2 O content (The number m is more than 4, Q n species indicate the GeO 4 tetrahedron with 4-n non-bridging oxygens and Q 3 and Q 2 are in Raman bands at 956 and 858 cm -1 for x=0.0 respectively).Q 3 and Q 2 in the m-membered chain of GeO 4 tetrahedra containing Q 3 or Q 2 is not broken down by K 2 O but reallocated to a metastable state by rapid-quenching.For 20 < x < 30, as 3-memebered rings of GeO 4 tetrahedra in the reallocated-and-isolated microstructure disappear, K 2 O breaks down other isolated structural unit, for example, 3-membered chain of GeO 4 tetrahedra and GeO 6 octahedra.For 30 ≤ x ≤ 40, the reallocated-main-network structure containing holes is also broken down to 3-membered chain of GeO 4 tetrahedra containing Q 3 or Q 2 by K 2 O.However, as 3-membered chain of GeO 4 tetrahedra containing Q 3 or Q 2 is included in the reallocated-and-isolated-microstructural units, the elastic modulus is also determined by the reallocated-and-isolated-microstructural units.Figure 6 shows the content dependence of integrated intensity of vibrational band centered at 520 cm -1 ascribed to 3-membered ring of GeO 4 tetrahedra (Mamiya et al., 2009).For x=0.0, there are 3-membered rings of GeO 4 tetrahedra in the reallocated-main-network structure containing holes and the vibrational band at 420 cm -1 drastically decreases with K 2 O content, whereas the vibrational band at 520 cm -1 drastically increases with K 2 O content.This means 6-membered rings of GeO 4 tetrahedra are converted to 3-membered rings of GeO 4 tetrahedra with K 2 O content.The e-membered rings of GeO 4 tetrahedra reach the maximum for about x=10 and drastically decrease, and then down to the initial value (for germania glass) for x=20, almost remain unchanged for 20 < x< 30, and afterward decrease again.Figure 7 exhibits the content dependence of Ge atom (Yiannopoulos et al., 1997).For x=0.0, every Ge atom is 4 coordination.When K 2 O is added to germania (GeO 2 ) glass, 6 coordinated Ge atom increases and reaches the maximum for about x=20 and then decreases down to the initial state for x= 45.This also indicates the number of GeO 6 octahedron.We investigate the change of microstructure with respect to the elastic modulus.For the purpose of description and discussion, it is useful to divide the range of K 2 O content into 4 regions, the low-K 2 O-content region from 0 to 10 mol%, the middle-K 2 O-content region from 10 to 20 mol%, the high-K 2 O-content region from 20 to 30 mol%, the higher-K 2 O-content region from 30 to 40 mol%.4.3.1The Low-K 2 O-Content Region (0 to 10 mol%) For x=0.0, there are a number of 6-membered rings of GeO 4 tetrahedra and a few m-membered chains of GeO 4 tetrahedra containing Q 3 or Q 2 .When K 2 O is added to the GeO 2 glass, 6-membered rings of GeO 4 tetrahedra are changed into 3-membered rings of GeO 4 tetrahedra and m-membered chains of GeO 4 tetrahedra containing Q 3 or Q 2 is converted into m-membered chains of GeO 4 tetrahedra and GeO 6 octahedron.Because the K 2 O breaks down 6-membered ring of GeO 4 tetrahedra into two pieces of 3-membered rings of GeO 4 tetrahedra and it also converts Q 3 or Q 2 to GeO 6 octahedron.The number of 3-membered rings of GeO 4 tetrahedra increases and the number of GeO 6 also increases.As the superposition of the two factors increases, the elastic modulus also increases.When we turn our attention to the vibrational intensity of the microstructure, the torque of 3-membered ring of GeO 4 tetrahedra is a quarter of that of 6-membered ring of GeO 4 tetrahedra, however, the number of 3-membered rings of GeO 4 tetrahedra becomes two times of that of 6-membered rings of GeO 4 tetrahedra.The torque means the moment of rotation around center of ring that is caused by asymmetric vibrational mode of Ge(4)-O-Ge(4).As the vibrational intensity of 3-membered rings of GeO 4 tetrahedra becomes a half of that of 6-membered rings of GeO 4 tetrahedra, the vibrational intensity drastically decreases.That is when the 6-membered ring of GeO 4 tetrahedra is changed into the two pieces of 3-membered rings of GeO 4 tetrahedra, its vibrational intensity decreases down to a half.In the case of m-membered chains of GeO 4 tetrahedra containing Q 3 or Q 2 , Q 3 or Q 2 has a large vibrational intensity because they contain non-bridging oxygens.Besides, GeO 6 octahedron has a small vibrational intensity because it is a stable state and a strong bonding of octahedron.That is when m-membered chain of GeO 4 tetrahedra containing Q 3 or Q 2 are conveted into m-membered chain of GeO 4 tetrahedra and GeO 6 , its vibrational intensity decreases.As the superposition of two factors drastically decreases, the vibrational intensity also drastically decreases.For x=10.0, 6-membered rings of GeO 4 tetrahedra almost disappear and the number of 3-membered rings of GeO 4 tetrahedra becomes the maximum.For 10.0 < x < 15.0, the number of 3-membered rings of GeO 4 tetrahedra decreases gradually and the number of GeO 6 increases drastically.As the superposition of the two factors still increases, the elastic modulus also increases.When we turn our attention to the vibrational intensity of the microstructure, 3-membered rings of GeO 4 tetrahedra are gradually broken down to 3-membered chains of GeO 4 containing Q 3 by K 2 O.As chain and Q 3 have a free end and non-bridging oxygen respectively, the vibrational intensity of 3-membered chain of GeO 4 tetrahedra containing Q 3 is larger than that of 3-membered ring of GeO 4 tetrahedra.When m-membered chains of GeO 4 tetrahedra containing Q 3 or Q 2 are drastically converted into m-membered of chains of GeO 4 tetrahedra and GeO 6 octahedra by K 2 O, the vibrational intensity decreases rapidly.As the superposition of two factors slightly decreases, the vibrational intensity of the two factors slightly decreases.For x=15, the elastic modulus is the maximum.On the contrary, the vibrational intensity is the minimum.For 15.0 < x < 20.0, 3-membered rings of GeO 4 tetrahedra keep decreasing drastically and GeO 6 octahedra gradually increase.As the superposition of the two factors slightly decreases, the elastic modulus slightly decreases.When we turn our attention to the vibrational intensity of the microstructure, 3-membered rings of GeO 4 tetrahedra keep being broken down to the 3-membered chains of GeO 4 tetrahedra containing Q 3 by K 2 O.The vibrational intensity of 3-membered ring of GeO 4 tetrahedra drastically increases, because of increasing species of Q 3 .The m-membered chains of GeO 4 tetrahedra containing Q 3 or Q 2 are slightly converted into GeO 6 octahedra, because the two bands in Raman spectra are very weak and Figure 7 shows the slope of the average coordination number of Ge atom (CN) is low.The vibrational intensity of GeO 6 octahedra slightly decreases.As the superposition of the two factors gradually increases, the vibrational intensity gradually increases.The number of 3-membered rings of GeO 4 tetrahedra almost remains unchanged.Because 3-membered ring of GeO 4 tetrahedra in the isolated structure is all broken down by K 2 O.We must consider 3-membered chain of GeO 4 tetrahedra containing Q 3 which 3-membered rings of GeO 4 tetrahedra were broken down to.As 3-membered chains of GeO 4 tetrahedra containing Q 3 are further broken down into 3-membered chaings of GeO 4 tetrahedra containing Q 3 and Q 2 , the elastic modulus drastically decreases.The number of the GeO 6 octahedra reaches the maximum for x=20 and then decreases.As GeO 6 octahedron is broken down to Q 3 by the K 2 O, the elastic modulus of GeO 6 octahedron begins to decrease.Therefore, the superposition of the two factors continues decreasing drastically and the elastic modulus of the glass continues decreasing drastically.When we turn our attention to the vibrational intensity, the number of 3-membered rings of GeO 4 tetrahedra almost remains intact.As the m-membered chains of GeO 4 tetrahedra and GeO 6 octahedra are converted into the m-membered chains of GeO 4 tetrahedra containing Q 3 , the vibrational intensity drastically increases.As the superposition of the two factors drastically increases, the vibrational intensity of glass drastically increases.As 3-membered chains of GeO 4 tetrahedra containing Q 3 and Q 2 are broken into 3-membered chains of GeO 4 tetrahedra containing Q 3 , Q 2 and Q 1 , the elastic modulus further decreases.The m-membered chains of GeO 4 tetraheda containing Q 3 are broken into the chains of GeO 4 tetrahedra containing Q 3 and Q 2 .When 3-membered rings of GeO 4 tetrahedra as the reallocated-and-isolated-structural unit disappear, 3-membered rings of GeO 4 tetrahedra in the reallocated-main-network structure are broken into the 3-membered chains of GeO 4 tetrahedra containing Q 3 by K 2 O. Raman bands at 866, 774 and 710 cm -1 indicate Q 3 , Q 2 and Q 1 (for x=39.0)respectively.These bands grow drastically.All the structures are broken down into the structures containing more Q 3 , Q 2 and Q 1 .Therefore, the elastic modulus decreases further drastically.When we turn our attention to the vibrational intensity, as the Q 3 , Q 2 and Q 1 have non-bridging oxygens respectively, the vibrational intensity of glass increases further drastically.

The Origin of the Excess Heat Capacity
The relationship between the elastic modulus and the vibrational intensity is dual.The change of the C p T -3 max is well explained by the change of the reallocated-and-isolated-microstructural units.And the hole model of liquid theory provides for resonant mechanism.Therefore, the origin of the excess heat capacity is the reallocated-main-network structure containing holes resonant with the reallocated-and-isolated-structural units.

Conclusion
The heat capacity C p of potassium germanate glass (xK 2 O•(100-x)GeO 2 ; x=0.0, 10.1, 19.0, 28.2, 39.0) was measured in the temperature range from 2 to 50 K.It was found that the reduced excess heat capacity C p T -3 is caused by resonance of the reallocated main network structure containing holes with the reallocated isolated microstructural units.The relationship between the reduced excess heat capacity C p T -3 max and the elastic modulus is dual.The C p T -3 max indicates degree of freedom in the reallocated-and-isolated microstructure in glass.The hole model of liquid theory is useful to explain the mechanism of producing the resonant structure in the glass.

Figure 3 .
Figure 3.The content dependence of maximum of reduced excess heat capacity C p T -3 max of potassium germanate glasses

Figure 4 .
Figure 4.The content dependence of elastic moluli of potassium germanate glasses