GibbsâThomson, Thermal GibbsâThomson, and HoffmanâWeeks

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Gibbs-Thomson, Thermal Gibbs-Thomson and HoffmanWeeks plots of polyethylene crystals examined by fastscan calorimetry and small-angle X-ray scattering Akihiko Toda, Ken Taguchi, and Koji Nozaki Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.9b00209 • Publication Date (Web): 13 Mar 2019 Downloaded from http://pubs.acs.org on March 21, 2019

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Crystal Growth & Design

Gibbs-Thomson, Thermal Gibbs-Thomson and Hoffman-Weeks plots

of polyethylene crystals examined by fast-scan calorimetry and small-angle X-ray scattering Akihiko Toda a*, Ken Taguchi a and Koji Nozaki b a

Graduate School of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan b

Graduate School of Sciences and Technology for Innovation, Yamaguchi University, Yamaguchi 753-8512, Japan

* Corresponding author, [email protected], Tel: +81-82-424-6558

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Crystal Growth & Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Abstract Crystallization and melting behaviors of polyethylene (PE) crystals were examined by fastscan chip sensor calorimetry (FSC) and small-angle X-ray scattering (SAXS).

The

melting point TM of chain-folded thin lamellar crystals of PE was determined by utilizing FSC, and the crystalline lamellar thickness dc by SAXS.

The range of crystallization

temperature Tc of SAXS as well as FSC was extended much broader than before by examining those samples prepared on a chip sensor of FSC and applied deep temperature jump for crystallization under large supercooling Tc.

While re-confirming the

equilibrium melting point TM0 with linear relationship of the Melting and Crystallization lines in the Gibbs-Thomson (G-T) plots of TM and Tc against (dc)-1, respectively, and the Hoffman-Weeks (H-W) plot of TM against Tc under relatively small Tc, those G-T and HW plots seriously deviated from linear straight lines under large Tc.

The origin of the

curved G-T plots was ascribed to the Tc dependent folding surface free energy e.

The

TM0 and Tc dependent e were convinced by a newly proposed Thermal G-T plot of TM against the inverse of specific heat of fusion hfs in terms of the increase during the secondary stage of crystallization on long time isothermal annealing at Tc of chain-folded PE crystals transforming to more stable states by lamellar thickening and crystal perfecting.

The deviation from linear H-W plot under large Tc was then ascribed to the

crystal reorganization on heating of those less stable crystals formed under large Tc even with fast heating. Application of fast heating and cooling by FSC plays an essential role in the confirmation of those thermodynamic behaviors of metastable polymer crystals with chain folding.


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Introduction On crystallization, linear-chain polymers are in a metastable crystalline state due to a characteristic manner of chain-folded crystallization forming quite thin lamellar crystals in the order of 10 nm (Fig. 1) [1,2].

The physical properties are then characterized by the

crystalline lamellar thickness dc and the melting point TM, which can be significantly lower than the equilibrium melting point of chain-extended infinite-size crystals, TM0 [3]. Owing to the metastability of those chain-folded polymer crystals, in order to avoid crystal reorganization on heating for TM determination, one should apply heating rate fast enough in the order of 1,000 Ks-1, which is now realizable by chip sensor calorimetry developed in recent years [4].

On the other hand, for those polymer crystals grown from the melt and

forming one dimensional stacks of crystalline and amorphous layers (Fig. 1), small angle x-ray scattering, SAXS, will be the best method to determine the layer thickness [5].

Figure 1. Schematic drawing of the one dimensional stack of crystalline and amorphous layers formed by crystalline polymers in a metastable state with chain folding.

Present paper examines the applicability of the characterization methods of crystalline polymers utilizing TM and dc obtained by those two methods of ultra-fast-scan calorimetry (FSC) with chip sensor and SAXS in the so-called Gibbs-Thomson (G-T) and Hoffman-Weeks (H-W) plots.

In addition to those conventional plots, we have recently

proposed a new different type of plot, Thermal G-T plot, utilizing data-sets obtained by only calorimetry [6].

As will be briefly summarized below, those three plots are basically

independent by utilizing different types of data-sets and applied to different stages of polymer crystallization. Examined polymer in the present paper is polyethylene, PE, the simplest linearchain polymer, which we have examined in a prior study within a range of crystallization temperature Tc limited by its fast crystallization rate [7].

In the analyses of PE, the

conventional G-T and H-W plots worked satisfactorily with TM determined by FSC. However, our subsequent examination of poly(vinylidene fluoride), PVDF, which 3 ACS Paragon Plus Environment


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crystallizes relatively slowly and hence could be examined over much broader Tc range, showed the failure of an ideal behavior of G-T plots, the origin of which was well understood by the interpretation deduced from the results of Thermal G-T plot [6].

In

order to confirm the general applicability of the mechanism found in those plots of PVDF, in the present paper, we re-examine those three plots of PE over much broader Tc range than before by utilizing the applicability of deep temperature jump of chip-sensor calorimetry not only for thermal analysis but for the sample preparation of SAXS, which is required to examine dc of those crystals formed under large Tc in a short time interval less than a second. In the following, we firstly give a brief summary of the determination methods of and TM and dc, and subsequently the analysis methods of those three plots, which are then followed by the experimental results and discussion.

Determination methods of TM and dc A. TM by FSC [7-9] We proposed to analyze the melting behavior of polymer crystals obtained by FSC under applied heating rate  by the following equation of the melting peak temperature Tpeak,

Tpeak  TM  A  z

(1)

with a coefficient A and a power z.

In general, the melting peak can be very broad due to

distributions of several factors, such as molar mass, crystalline thickness and defects, and due to crystal reorganization including possible melting-recrystallization-melting process of those metastable chain-folded polymer crystals.

By applying heating rate fast enough,

we can avoid the secondary process of re-organization, and then we can suppose that Tpeak represents the melting points of the majority of crystallites formed under crystallization conditions.

In eq. (1), we have shown that the second term is introduced by the melting

kinetics under superheated state and the thermal lags of fast heating, which can be experimentally evaluated by the plot of Tpeak against  z with the power z utilized as an adjustable parameter for the linear fitting, as shown in Fig. 2a. Then, TM is determined as the y-axis intercept of the fitting linear straight line and represents the melting point at zero-heating-rate, which is the melting point of chain-folded crystals under equilibrium with surrounding melt.

This TM is then utilized for the determination of TM0 by the above

mentioned three plots.

TM0 is the most important parameter which determines thermal 4 ACS Paragon Plus Environment

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properties of crystalline polymers, but there is no straightforward determination method due to the formation of metastable crystals with chain folding.

For example, due to

crystal reorganization on slow heating close to zero heating rate, Tpeak can increase with slower  , as shown in Fig. 2a, so that the application of heating rate fast enough and the extrapolation of those data to zero heating rate are inevitable for the TM determination of metastable chain-folded polymer crystals. (a)

(b)

K(x)/K(0)

1

Tpeak

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


L = dc+da 0

TM 0

(/0) z

x /nm

d0

Figure 2. Schematic drawing of (a) Tpeak( z) and (b) K(x).

B. dc by SAXS Following the standard method [5] applying Fourier transform to the Lorentz-corrected scattering profile of small angle X-ray scattering, SAXS, we can determine the onedimensional auto-correlation function K(x), which is then used for the thickness determination of the one dimensional stacks of crystalline and amorphous layers: 

K ( x)   2 s 2 I ( s ) e2πi sz ds

(2)



with the scattering intensity I(s), the magnitude of a scattering wave vector s = (/2) sin , the wave length of X-ray  and the scattering angle 2.

Fig. 2b shows a schematic K(x)

with d0 = dc or da, crystalline or amorphous layer thickness, respectively, and the long spacing L= dc+da.

For PE, dc is thicker, while larger thermal expansion of da can be

utilized for the identification of d0 of unknown polymers.

Thermodynamic relationships of chain-folded polymer crystals For chain-folded polymer crystals with crystalline lamellar thickness dc in nm order, on the basis of the Gibbs-Thomson effect of small systems, the following relationships hold.



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A. Melting and Crystallization Lines in the G-T plots [2,3,5]: With the folding surface free energy e and the heat of fusion per unit volume hf, C dc C Tc  TM0   dc

TM  TM0 

Melting line

(3)

Crystallization line

(4)

2 eTM0 C h f

(5)

Here, the expression of the Crystallization line assumes a constant thickening factor  after the formation of a critical surface nucleus on the growth face with the critical thickness of dc*, which applies to PVDF [6] and PE [7].

The original expression of dc satisfied at Tc is

given as follows, d c   d c*  

C T  Tc

(6)

0 M

B. H-W plot [2,3,10,11]:

TM  TM0 

1



(Tc  TM0 )

(7)

This H-W relationship is derived by combining those two expressions of the Melting and Crystallization lines of eqs. (3) and (4) having the common factor C/dc under the assumption of unchanged coefficient C (and e) and thickness dc on melting at Tpeak from those at Tc.

This condition supposes the melting on heating at a rate fast enough to

guarantee no lamellar thickening and unchanged structure of folding surface during the heating process. C. Thermal G-T plot [6]: After the completion of the primary stage of crystallization by filling-in the sample space with spherulitic growth of chain-folded lamellar crystallites of polymers, because of their metastability, those crystallites undergo further crystallization as the secondary stage by crystal thickening and perfecting, both of which bring the increase in TM and the total heat of fusion Hf. [12-15].

Then, the plot of TM against 1/Hf of the data-set obtained under

isothermal condition in the secondary stage will be on a linear straight line because hf dc in eq. (3) will be replaced by Hf /Ac with Ac representing the total surface area of those lamellar crystals which are formed in the primary stage and will keep constant Ac during the secondary stage: 6 ACS Paragon Plus Environment

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TM  TM0 

2 eTM0 Ac C  TM0  H f hfs

(8)

2 eTM0 (9) L Here, in eq. (8), the total Hf is replaced by the specific hfs = Hf /Vsample while the total C 

volume of sample Vsample is represented by the product of Ac and long spacing L as Vsample = LAc.

Because this plot is based on the G-T plot of eq. (3) and both of TM and Hf are

available with only calorimetry, we name this plot Thermal G-T plot.

Evaluation methods of TM0 and e Those three different plots mentioned above are independent from each other by utilizing different data-sets (Tc, TM), (dc-1, TM) and (Hf-1, TM) at different stages of crystallization. They can be utilized for the evaluation of TM0 and e. A. TM0 The equilibrium melting point TM0 can be independently determined by those three plots as the y-axis intercepts of the fitting linear straight lines. B. e Folding surface free energy e can be evaluated from the slopes of the fitting linear straight lines of the original and thermal G-T plots independently, namely from the coefficient C and C' of eqs. (5) and (9), by utilizing hf of perfect crystals taken from literature and L from the experimental data of SAXS, respectively.

Our Prior results of PE [7] and PVDF [6] with FSC and SAXS Figure 3 shows the H-W, G-T and Thermal G-T plots of our prior experimental results of PE and PVDF by using FSC and SAXS.

Firstly, with PE, both of the H-W and G-T plots

are on linear straight lines, as schematically shown in Figs. 3a and 3b.

The y-axis

intercepts are in good agreement, confirming the applicability of those plots for TM0 determination.

Secondly, with relatively slow crystallization rate of PVDF, the examined

Tc range became broader, and the H-W plot started to deviate from linear relationship under large Tc at low Tc, as shown in Fig 3c. On the other hand, the G-T plots of both of Melting and Crystallization lines in Fig. 3d seriously deviated from straight lines even in The TM0 determined by the H-W plot could

the Tc range above Tc1 with linear H-W plot. 7

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not be justified in the G-T plots.

Page 8 of 24

Then, the Thermal G-T plots in Fig. 1e were on straight

lines at respective Tc, with the y-axis intercept in common with the H-W plot; the slopes of those straight lines of isothermal data-sets were dependent on Tc.

The interpretation of

the curved H-W and G-T plots and the Tc dependent linear Thermal G-T plot will be discussed in conjunction with the present results of PE re-examined for much broader Tc range. (a) H-W (PE)

TM0

(b) G-T (PE) TM

TM

TM

Tc

T

(dc ) 1

Tc (c) H-W (PVDF)

TM0

(d) G-T (PVDF) TM

TM0

TM

TM

TM

(Tc )1

Tc

T

(e) Thermal G-T (PVDF)

TM

( Tc )2

Tc1

( Tc )3

Tc1

(dc ) 1

Tc

(Hf ) 1

Figure 3. Schematic drawing of our prior results in terms of (a, c) H-W, (b, d) G-T and (e) Thermal G-T plots: (a, b) PE [7] and (c-e) PVDF [6].

Experimental Linear polyethylene, PE (SRM1475, Mw=5.20×104, Mw/Mn=2.90), was purchased from NIST (Gaithersburg, USA). The crystallization and melting were examined by fast-scan FlashDSC1 (Mettler-Toledo) with microchip sensor, UFS1 with and without refrigerated cooling system. cell.

Dry nitrogen gas with a flow rate of 30 mL min-1 was purged through the

Figure 4 shows typical film samples prepared with a microtome (Leica ultracut

UCT, Austria) from a pellet.

Isothermal crystallization under large Tc was done with

chiller and checked by the time evolution of the exothermic heat flow; the Tc dependence 8 ACS Paragon Plus Environment

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of peak time was in good agreement down to Tc=89.3oC with that of a high density polyethylene recently reported in literature [16].

The temperature of each chip sensor was

calibrated by the melting onset temperature of indium (99.999 %, Niraco, Japan) at  = 10 K s-1.

Figure 4. Optical microscopic images of PE thin films on chip sensors: (a) 1 m thick film for thermal analysis and (b) 5 m thick film prepared for SAXS.

Measurements of small angle X-ray scattering, SAXS were done with NANOViewer (Rigaku) with Cu-Ka radiation (40 kV and 30 mA, =0.154 nm). The camera length was 940 or 1,227 mm. In-situ measurements on crystallization were done for Tc116.5oC at relatively small Tc with a home-made three sets of hot cells connected in series to apply temperature jumps by dropping sample from the hot cell at the top (constructed by NISSIN SEIKI Co., Ltd., Japan); the details are given in ref. [6].

The

sample thickness was 200 m, and the exposure time was 3-10 min. Ex-situ measurements were also done for the samples crystallized on a chip sensor of FSC, as shown in Fig. 4b, and those prepared by a temperature jump on dropping in a hot water bath set at Tc.

Sample thickness of those on chip sensor was 2, 5 and 15 m, and the

exposure time was 2-4 h.

Those samples prepared in a hot water bath were 20 and 30 m

thick films backed by a capton film of 12.5 m in thickness and supported by a circular copper plate of 0.1 mm in thickness with a hole of 3 mm in diameter; the exposure time was 40 min.

The ex-situ SAXS measurements were done for those samples placed on a

hot plate kept at about 50oC.



Results FSC

HF /mW

0

o

Tc=112.6 C t=0.6 s  dep.

endo.

-1

(a)

120

140

160

0

-0.4

o

endo.

HF /mW

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 24

Tc=112.6 C t dep. -1. =1,000 Ks

(b)

120

130

140

o

T/C

Figure 5. Melting peaks of PE crystals formed by the indicated conditions.

Figures 5 shows the typical examples of the endothermic melting peak of PE crystals prepared under indicated conditions.

The heating rate dependence of the peak

temperature Tpeak of those endothermic peaks in Fig. 5a is analyzed in a quantitative manner on the basis of eq. (1).

The details of the analysis were discussed in a separate

paper [17] with the basic trend of the adjusted power z being dependent on both of Tc and t: smaller z at higher Tc and longer t.

In our prior studies, we have shown that, for film

samples thin and small enough, thermal lags have negligible influence on the power z, which is then determined by the melting kinetics under superheated state; basically, 1 m thick film of 100100 m2 in area is thin and small enough to reduce thermal lags for

 Tc2 suggest the

leveling-off of e at high Tc under small Tc. As schematically shown in Fig. 12c, the present result of the Crystallization line in Fig. 12b can be replotted in terms of the Tc dependence of dc.

We can explain those

behaviors of dc if we suppose the following Tc dependence of e,

0e e (Tc )   0 e [1  yc (Tc2  Tc )] with the coefficient yc > 0.

for Tc  Tc2 for Tc  Tc2

(10)

Then, from eqs. (5) and (6), the behavior in Fig. 12c will be

represented as, 16 ACS Paragon Plus Environment

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 1 2T   Tc dc   dc*    h f 1  yc Tc2  y c  Tc 0 M

for Tc  Tc2

0 e

(11) for Tc  Tc2

The same constant term yc was required for the Tc dependence of solution-grown crystals, though in the Hoffman's argument the thickening coefficient was supposed to be unity because solution-grown crystals were believed to undergo no thickening.

Figure 13

shows the Crystallization line and the Tc dependence of dc over Tc range broader than in Fig. 11b by including the data of those samples prepared in a hot water bath under large Tc; the classical data of quenched droplet samples taken by Barham et al. [26] are also It is seen that similar trend of Tc dependence holds over the broad

plotted for comparison. Tc range. 140

30

(a)

120

20

dc /nm

o

(b)

Tc

130

Tc / C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


in situ 200m

110

FSC 15m FSC 5m FSC 2m bath 20m

100 90

10

Barham et al. [19]

80 0

0.05

(dc/nm)

0

0.10

0

0.02

-1

0.04

o

(Tc/ C)

0.06 -1

0.08

Figure 13. (a) Crystallization line in the G-T plot and (b) corresponding Tc dependence of dc.

In the final part of the discussion on the H-W and G-T plots, in terms of the linear relationship of the H-W plot in the same Tc range above Tc1 as of the curved G-T plots, the linear relationship will be retained under the condition of unchanged coefficient C (e) and thickness dc on melting at TM (Tpeak) from those at Tc, as mentioned in the derivation of the linear H-W relationship of eq. (7) from eqs. (3) and (4) because the common factor C/dc is then canceled out in the H-W plot.

Therefore, the curved G-T plots and the linear H-W

plots are in accordance with each other if the curved G-T plots are caused by this Tc dependent e and C.

Here, the condition of unchanged C (e) and dc supposes the melting

on heating at a rate fast enough to guarantee no crystal reorganization during the heating



process.

Therefore, the deviation from linear relation below Tc1 in Figs. 11a and 12a will

most probably be due to the reorganization of metastable chain-folded polymer crystals on heating even with fast scan by FSC for those less stable crystals formed at lower Tc with thin lamellar thickness and lower crystal perfection.

There can be two different types of

reorganization which bring upward deviation of TM in the H-W plot at lower Tc with increasing distance between Tc and TM. 1. Crystal reorganization to more stable crystals with thickening and perfecting, which brings higher TM. 2. Reorganization of folding surface, which brings e(TM) < e(Tc), i.e. decoupling of C of the Melting line form that of Crystallization line C(TM) < C(Tc) in eqs. (3) and (4), respectively, and hence less steep slope of the H-W plot, as expressed in the following, TM  TM0 

1 C (TM ) (Tc  TM0 )  C (Tc )

(14)

It is noted that eq. (14) suggests Tc dependent slope of consequently curved H-W plot and was the basis of the non-linear H-W plots proposed by Marand et al. [27].

Thermal G-T plot applied to the secondary crystallization 140

135

o

TM / C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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130

o

Tc / C 127.6r 117.6 107.6

125

0

122.6 112.6 102.6 2 

4 -3 -1

6

(hfs0 J m )

Figure 14. Thermal G-T plot of TM against (hfs)-1 for the t range in the secondary stage of isothermal crystallization. The straight lines are drawn from the fixed TM0 = 141oC. At the highest Tc, crystals were formed by recrystallization of once molten crystals formed by cooling in order to shorten the completion time of primary stage.

Figure 14 shows the Thermal G-T plots applied to the secondary stage of isothermal crystallization; the crystallization time t was 1-20,000 s depending on Tc.

They are on

linear straight lines at respective Tc by using the fixed y-axis intercept TM0 determined by 18 ACS Paragon Plus Environment

Page 19 of 24

the H-W and original G-T plots in Fig. 11.

The slopes of those straight lines of isothermal

data are dependent on Tc, suggesting the Tc dependence of the slope C' in eq. (8) and hence of e in eq. (9) in accordance with the explanation for the curved G-T plots. 100

-2

80

e /erg cm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


60

Tc2

Tc1 40 20 0

G-T(TM vs dc-1) G-T(Tc vs dc-1) thermal-G-T(TM vs hfs-1) 100

110

120

130

o

Tc / C

Figure 15. Tc dependence ofe evaluated by the G-T and Thermal G-T plots of Figs. 11b and 14, respectively.

The consistence is further convinced by the comparison of e determined from the slopes of the Melting line of the G-T plot and the Thermal G-T plot on the basis of eqs. (4) and (5) and eqs. (8) and (9), respectively, as summarized in Introduction.

For PE, the

literature value of hf = 2.8109 erg cm-3 [2] was employed in eq. (5), and the values of long spacing L in eq. (9) were taken from Fig. 10; namely, we neglected the change in L with t, but the change was less than 1 nm for t up to 20,000 s and much smaller than that of dc shown in Fig. 9.

Figure 15 shows a good agreement between those evaluated values

above Tc1; e was evaluated for three different samples with the Thermal G-T plot. Figure 15 also shows the e evaluated from the Crystallization line of the G-T plot on the basis of eq. (4) with  =1.74 determined by the H-W plot; the values are also in good agreement above Tc1.

The disagreement below Tc1 has the same meaning as the deviation

from linear H-W plot below Tc1 in Fig. 11a because the H-W, the Melting line of G-T and the Thermal G-T plots utilize TM determined by the melting on heating while the Crystallization line is the plot of Tc in the G-T plots.

From the slope of the fitting line in

Tc1

GibbsâThomson, Thermal GibbsâThomson, and HoffmanâWeeks

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