MD Karin Hedenmalm
email: karin.hedenmalm@mpa.se

MSc Annika Johansson

MSc Kristoffer Bäckman

Prematurely ended phase III trials in Sweden during the years 2002-2008 by

Keywords: clinical trials phase III, data monitoring committee, safety, efficacy, premature ending

Purpose The present investigation is aimed at identifying prematurely ended phase III clinical trials (CTs) and their proportion of all phase III CTs, reviews reasons for the premature ending, examines if a data monitoring committee (DMC) was involved in the decision, identifies the data source on which the decision was based and reviews consequences of the premature ending for product development. A further aim is to identify risk factors for a premature ending.

Methods Prematurely ended phase III CTs in Sweden during the years 2002-2008

were identified by database searches. Identified trials were reviewed with regard to tested treatment, study design, reasons for the premature ending, data source, on which the decision was based, and existence of and recommendation from a DMC.

Three randomly selected not prematurely ended control trials were identified matched on application year starting 1 May 2004.

Results A total of 84 phase III CT applications (8%), were prematurely ended. Most trials were ended due to safety and/or efficacy concerns. A DMC was more common among trials investigating mortality as primary endpoint and oncology trials. A recommendation from the DMC to terminate the trial was most likely in case of combined safety and efficacy related issues arising from within the trial. Possible risk factors for a premature ending included mortality as an endpoint, obesity as indication and a longer planned study duration. Around 30% of trials with active substances without a marketing authorisation at the time of the clinical trial application were discontinued from further development.

Conclusions DMCs in phase III CTs are used in accordance with guidelines. Use of DMCs was associated with possible risk factors for a premature ending. Use of DMCs was numerically, but not significantly associated with a premature ending.

Background

Phase III is the last phase of the drug development process [START_REF]ICH E8 Note for guidance on general considerations for clinical trials[END_REF] before a new marketing authorization application can be submitted. Phase III clinical trials are intended to verify the clinical efficacy and characterize the safety profile of a drug in a therapeutic indication. Whilst it is important that such trials are not ended prematurely before efficacy has been unequivocally demonstrated it is equally necessary to avoid continued exposure to a harmful drug. In order to accommodate these needs without risking the scientific integrity of the trial, the review of data from ongoing clinical trials is usually delegated to an independent external committee or advisory board which makes recommendations to the sponsor of the trial [START_REF]ICH E6 (R1) Note for guidance on good clinical practice[END_REF]. Recently, both the EU and the FDA have published guidelines on the use of such committees in clinical trials [START_REF]Guideline on data monitoring committees[END_REF][START_REF]Guidance for clinical trial sponsors. Establishment and operation of clinical trial data monitoring committees[END_REF].

For consistency, the term data monitoring committee (DMC) will be used throughout this document.

The aim of this investigation was to identify phase III clinical trials which were ended prematurely in Sweden in the years 2002-2008, to review the reasons for the premature discontinuation of the trial, to find out whether or not a DMC had been involved in the decision and if the data on which the decision to stop the trial came from the trial itself or from outside of the trial. Further aims were to identify the proportion of phase III clinical trials that were prematurely ended and to review the consequences of the premature ending for the tested product and its indication.

Methods

In Sweden, approval from the Medical Products Agency (MPA) has been obligatory for all clinical trials involving human subjects since 1988. Limited administrative information related to the clinical trials is stored in the databases LVIS-C (for clinical trials before 1 May 2004) and Documentum (for clinical trials from 1 May 2004). The European Clinical Trial Directive (EU-CTD) was introduced for all clinical trial applications since 1 May 2004. The European database EudraCT also contains administrative information. Information that can be found in the databases include e.g. application date, title of the trial, trial phase (only LVIS-C and EudraCT), name/s of the investigational medicinal product/s, date/s of request/s for supplementary information, name of sponsor, name/s of investigators, date/s when the trial was approved or rejected, and status of the trial (approved, interrupted, completed).

Phase III clinical trials that were prematurely ended between 1 January 2002 and 31 December 2008 were identified in LVIS-C, Documentum and EudraCT. Since trial phase is not included in Documentum, all applications for phase III clinical trials in Sweden since 1 May 2004 were identified in EudraCT and records were matched with the same records from Documentum. In the databases LVIS-C and Documentum, the current status of the trial, i.e. "approved", "interrupted" or "completed" is indicated. All trials with the status "interrupted" were reviewed and trials that were permanently ended were included in the study. In addition, since the status of a trial changes to "completed" after the final study report has been submitted, the EudraCT database was searched for the term "premature end" and for reasons for a "premature end" (these reasons are "safety", "lack of efficacy", "the trial has not commenced" and "other"), and the LVIS-C database was searched for spontaneous notifications using loose-ending text strings (indicated by *) that may indicate a premature ending ("stop*", "avbr*", "prematu*", "förtid*" and "säkerhet*). All trials retrieved by the searches were reviewed in order to identify trials that were permanently and prematurely ended.

The review was restricted to trials that were both approved and had started patient inclusion in Sweden. A prematurely ended trial was defined as a trial that was ended prior to the last treatment of the last patient according to the last version of the protocol. The reasons for a premature ending were classified as due to safety alone, efficacy alone, both safety and efficacy, and other reasons including e.g. slow patient recruitment, a lower than expected endpoint rate, changes to regulatory guidelines in such a way that the trial hypothesis was no longer of interest etc. A safety issue was considered in case of an overrepresentation of serious adverse events or reactions, even if not formally statistically significant, provided that the pharmacological mechanism of action made a causal relationship likely, and provided that the safety issue was not outweighed by other advantages. A safety issue was also considered if the tested treatment had the opposite to the expected effect on the primary endpoint in the absence of any benefit. A smaller than expected effect or the opposite effect on the primary endpoint was considered a lack of efficacy issue. A greater than expected effect on the primary endpoint was considered an unexpected positive efficacy finding. The categorization was based on an assessment of the overall profile of the tested treatment.

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All prematurely ended trials were further reviewed with regard to the tested treatment, the type of control group used in the trial (placebo, active treatment, another dose or regime of the study treatment, background, standard or no treatment or no control group), and the existence of a DMC. It was noted if the primary endpoints concerned mortality.

For clinical trial applications during the years 2002 to 2006 the proportion of all phase III trials that were prematurely ended until 31 December 2008 was calculated. This was not done for clinical trial applications during the years 2007-2008 due to an insufficient period of follow-up that was considered likely to result in a significant number of prematurely ended trials in the years to come.

In order to evaluate the impact of the premature ending on the study duration and to identify possible risk factors for a premature ending, a total of three randomly selected phase III control trials that were not prematurely ended until 31 December 2008 were identified for each prematurely ended trial. The control trials were matched on application year. The comparison was restricted to clinical trial applications between 1 May 2004 and 31 December 2008 due to difficulties in obtaining information from control trials before the introduction of the EU-CTD.

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Fisher's exact tests and the Mann Whitney U-test were performed to compare groups of interest. A two-sided p value of <0.05 was considered statistically significant.

Results

A total of 84 prematurely ended phase III clinical trials were identified during the years 2002-2008, see Figure 1. The trials had started up to 7 years before the premature ending, see Table 1. Almost all premature endings had taken place within the first three years after start of the trial (77/84; 92%) with the highest observed frequency during the second year (38/84; 45%). The reasons for the premature ending were safety alone in 28 cases (33%), safety and lack of efficacy in 25 cases (30%), lack of efficacy alone in 12 cases (14%) and other reasons in 19 cases (23 %).

The percentage of prematurely ended phase III trials in relation to the total number of phase III clinical trial applications during the years 2002-2006 is shown in Table 2.

Oncology was the largest treatment indication among prematurely ended phase III trials (24%; 20/84). Mortality was included as a primary endpoint in 29 trials (35%).

A total of 57 prematurely ended phase III clinical trials had a DMC. The distribution of the existence of a DMC in relation to different factors is shown in Table 3. In a total of 30 trials the DMC recommended its termination. This decision was based on safety alone, lack of efficacy alone or on both safety and lack of efficacy in all except one trial. For the distribution of reasons for termination (excluding reasons for termination other than safety and/or efficacy) in relation to the existence of a DMC, and in relation to the data source upon which the decision was based, see Table 4. No trial was ended prematurely due to unexpected positive benefit.

Compared with control trials, factors that appeared to be associated with a premature ending included treatment indication, mortality as an endpoint and planned study duration. Obesity as an indication was more common (10/48 vs 2/144; p<0.0001) whereas unusual indications (with fewer than five trials in our material) were less common (27/144 vs 2/48; p=0.02) among prematurely ended clinical trials. Mortality as an endpoint was twice as common (33%; 16/48 vs 17%; 24/144; p=0.02) and the planned study duration was slightly longer among prematurely ended clinical trials (median 30 months; 25 th to 75 th percentile 24-42 months vs 24 months; 25 th to 75 th percentile 14-46 months; p=0.03). For other factors no significant differences could be observed including design of the study, existence of a DMC or active substance with or without a marketing authorisation at the start of the study, although the existence of a DMC was numerically higher among prematurely ended trials (71%; 34/48 vs 56%; 80/144; p=0.06). Situations when a DMC was used were similar among prematurely ended trials and control trials with no significant differences observed. Despite the longer planned study duration, the real study duration was shorter among prematurely ended clinical trials, see Figure 3. Control trials that were still ongoing at the end of ourstudy were censored.

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A total of 36 trials (42%) concerned substances that were approved as medicinal products at the time of the clinical trial application, and 49 trials (58%) concerned substances that had not yet been approved at the time of the clinical trial application.

A total of 6 trials with already approved substances (17%; 6/36) were prematurely terminated following the withdrawal of the product from the market due to safety issues (n=3 for rimonabant, n=2 for rofecoxib, and n=1 for melagatran/ximelagatran).

In five trials with already approved substances (14%; 5/36) within unapproved treatment indications, a decision to discontinue pursuing the indication appeared to have been taken as a result of the trial; prevention of sporadic colorectal polyps in patients without a history of familial adenomatous polyposis (n=1 for celecoxib), prevention of major coronary events in postmenopausal women (n=1 for raloxifen), prevention of diabetic retinopathy (n=1 for sandostatin), treatment of climacteric symptoms in patients with a history of breast cancer (n=1 for tibolone), and treatment of non-haemophiliac patients with bleeding refractory to standard treatment following major trauma (n=1 for recombinant factor VIIa). Out of 49 trials with substances that

had not yet been approved at the time of the clinical trial application there were two trials (4%; 2/49) with substances that later became approved and were withdrawn from the market due to safety issues (n=1 rimonabant, n=1 rofecoxib). The phase III trials were prematurely ended after the market withdrawals. A further 16 trials (33%; 16/49) concerned 10 substances that were discontinued prior to the application for a marketing authorisation.

Discussion

During a 7-year-period a total of 85 prematurely ended phase III clinical trials were identified with a surprisingly large number in 2008 (n=35; 41% vs 14% expected). The withdrawal of rimonabant and other cannabinoid antagonists for obesity importantly contributed to the number of premature endings in 2008, accounting for a total of 10 prematurely ended trials that year.

The use of a DMC seemed to adhere with recommendations in the DMC guidelines as all trials investigating mortality as a primary endpoint had a DMC (29 prematurely ended trials and 24 control trials) as did all trials investigating cannabinoid antagonists due to the psychiatric adverse event profile of these agents (10 prematurely ended trials and two control trials), and all except one oncology trial (all 20 prematurely ended trials and 24/25 control trials). Overall, the use of a DMC was quite high, encompassing as much as 71% of prematurely ended phase III trials and 11(24) 56% of control trials with an application date from 1 May 2004 and onward. As expected, a DMC was less common in trials that had no control group.

In trials that had a DMC, the DMC was more likely to recommend termination of the trial for combined safety and lack of efficacy issues compared with safety-alone or lack of efficacy-alone related issues. Among trials that had a DMC that were stopped due to lack of efficacy-only issues the recommendation to stop the trial was more likely to be made by the DMC if data arose from the trial itself than if data arose from outside of the trial. The same tendency was also noted for safety issues. A possible explanation for this finding may be that DMCs are not always completely informed of results from other ongoing or recently completed trials with the tested product. Another possible explanation is that DMCs may be more focused on information from within the trial compared with information from outside the trial. Preferably, DMCs should be continuously informed of all results of relevance for their ongoing assessment. The one trial that was stopped for combined safety and efficacy issues without the recommendation of a DMC deserves some attention. The sponsor had analysed the data and made their decision before the DMC was informed; a procedure which puts the role of the DMC somewhat in doubt. In all instances, the sponsor adhered to the recommendations of the DMC.

Out of 29 trials that were recommended to be stopped by the DMC, 15 were stopped as a consequence of having fulfilled pre-specified criteria for futility at interim analysis and 14 trials were stopped without having fulfilled such criteria. No interim analysis was planned in 7 of those 14 trials. In the remaining 7 trials an interim analysis was planned. However, in two of the 7 cases the evaluation was triggered by results from other trials, and hence did not take place at the planned time of the interim analysis. In all 7 cases the DMC recommended to stop the trial even though formal statistical criteria were not met.

It was difficult to find out what impact, if any, the prematurely terminated trial had on the development of the tested substance. However, at least 30% of trials with active substances that did not have a marketing authorisation at the start of the trial were discontinued from further development. Even 19% of trials with marketed substances were prematurely ended due to withdrawal of the substance from the market for safety reasons. Among trials with approved substances, ten (31%) were stopped for safety reasons without a subsequent withdrawal of the marketed product. Marketing authorisation status did not appear to be associated with the risk of a premature ending as the proportion of trials with active substances with and without a marketing authorisation was similar among prematurely ended and control trials. Surprisingly, we did not identify any trials that were ended prematurely due to favourable results, which may be in contrast with findings from previous publications [START_REF] Trotta | Stopping a trial early in oncology: for patients or for industry?[END_REF][START_REF] Montori | Randomized trials stopped early for benefit: a systematic review[END_REF]. However, it is our genuine experience that stopping a trial for benefit is extremely rare. In the publication by Montori et al, only 143 out of 58357 published trials (0.2% or 2 per 1000) were stopped for benefit. The limited number of phase III trials investigated in this study may therefore explain why no such trial was identified, taking also into account biases associated with the publication of clinical trials. It is also possible that agreed protocol changes are made in case of trials with unexpected benefit in such a way that a premature ending according to our definition can be avoided. This possibility is supported by another publication of early reporting of positive trial results by Korn et al., where follow-up results were later published for 22/27 trials (81%) [START_REF] Korn | Stopping or reporting early for positive results in randomized clinical trials: the National Cancer Institute Cooperative Group experience from 1990 to 2005[END_REF]. We did identify two of the 25 trials reported to have been 13(24) stopped early for benefit by Trotta et al. In both cases beneficial interim results had been published. The trials, which were both open-label, were continued in accordance with the protocol. They did, therefore, not fulfil our definition of a premature ending. None of the trials reported by Korn et al. had been carried out in Sweden. It may be noted that according to the new EU guidance document related to the application of clinical trials with medicinal products for human use that was published in the spring of 2010, only premature endings related to safety concerns have to be reported.

Our study has a number of limitations. Firstly, the study was carried out only in Sweden during a limited time period, and results may not be representative for other countries or other periods of time. Secondly, the study has an exploratory character and no corrections for multiple comparisons have been made in the analyses. Some of the findings in the study may hence have arisen purely by chance. Finally, we were only able to compare prematurely ended trials with control trials for trials that had an application date of 1 May 2004 or later. For this reason our study had limited power to study factors that might be related to the risk of a premature ending. The size of the control group in our study (three controls per case) was selected bearing in mind that the incremental increase in statistical power decreases for each added control per case. Almost no further gain in power is achieved with ≥ 4 controls per case. In relation to the achieved increase in power three control trials for each prematurely ended trial seemed to adequately balance the workload.

Conclusion

Five to 14% of phase III clinical trials that had been submitted to the Medical Products Agency during each of the years 2002 to 2006 were prematurely ended.

Most trials were ended due to safety and/or efficacy concerns. No trial was ended due to unexpected favourable results. A DMC was used in two-thirds of prematurely ended trials and over half of control trials that were not prematurely ended with no statistically significant difference between the groups. The use of a DMC was more common among trials investigating mortality as primary endpoint and oncology trials.

The DMC was more likely to recommend terminating the trial for combined safety and efficacy related issues compared with safety-alone or efficacy-alone related issues.

Combined safety and efficacy related issues, in contrast to safety-alone or efficacyalone related issues, were mainly based on information from the trial itself.

Compared with phase III trials that were not prematurely ended, factors that were overrepresented among prematurely ended trials included mortality as endpoint, obesity as indication and planned longer study duration. Unusual indications (with fewer than five trials in our material) were in contrast underrepresented among prematurely ended trials. Despite anticipated longer study durations for prematurely ended trials, actual study durations were shorter. Around 30% of prematurely ended trials with active substances that did not have a marketing authorisation at the time of the clinical trial application resulted in the discontinuation of further development of the substance.

  The peak for premature endings took place about two years after start of the trial and the largest proportion of prematurely ended phase III trials concerned trials that started in 2006 (14% vs 5-10% in the years 2002-2005). In order to study if an increased number of premature endings may have been related to an increased use of DMCs recommended by the guidelines on the use of DMCs in clinical trials by EMEA and the FDA that came into effect in 2006, the use of DMCs among prematurely ended and control trials before (1 May 2004 until 31 December 2005) and since 1 January 2006 was compared. No increased use of DMCs was observed.

Figure legends:Figure 1 :Figure 2 :

 12 Figure legends:

Table 2 . Percentage of prematurely ended phase III trials in relation to the total number of phase III trial applications Total no. of applications for phase III trials No. of prematurely ended phase III trials

 2

	Percentage of
	prematurely ended
	phase III trials

Table 3 . The existence of a Data Monitoring Committee (DMC) in prematurely ended phase III clinical trials in relation to different factors

 3

	Total no	No. of	Percent of	P value 1
	of trials	trials	trials with
		with DMC	DMC

Funding acknowledgement: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors


Pottegard and Hallas, Physicians' and pharmacies' overview of the

Introduction

The risk of adverse drug reactions, polypharmacy, drug interactions and unintentional co-prescribing has increasingly become a problem, following the rise in intake of medicine (1;14). Many interventions aim to decrease these adverse events. To do so, an overview of the individual patient's medicine intake is necessary. However, several studies have revealed enormous discrepancies between the general practitioner's records, hospital admission papers, pharmacy records and the patient's own medicine cabinet (2; 3; 4; 5; 6; 7; 8; 9). It has been shown, among elder patients, that the number of prescribing physicians is an independent risk factor for experiencing an adverse drug event (11; 12; 6). Already in 1987 Gilchrist et al. showed up to two thirds of patient's drug history obtained from the general practitioner to be inaccurate [START_REF] Gilchrist | Prospective study of drug reporting by general practitioners for an elderly population referred to a geriatric service[END_REF]. It has since then repeatedly been proven that GP records (5; 6; 8; 2; 4) as well as hospital records (3; 4) and even patient reporting (5; 7; 3; 4) shows major discrepancies when compared to more thorough medication reviews, with up to 25 % of prescribed drugs being used without the general practitioners knowledge [START_REF] Barat | The consumption of drugs by 75-year-old individuals living in their own homes[END_REF]. A Danish study suggests that the use of a nationwide database may prove to be the most accurate measure of actual drug use [START_REF] Glintborg | The use of nationwide on-line prescription records improves the drug history in hospitalized patients[END_REF].

The two central players in this field are the prescriber and the pharmacy. We attempted to assess their overview by the 'Fidelity Coefficient', a measure of what proportion of individual patients' medication that are accounted for by their most used prescriber and pharmacy.

Materials and setting

The data for this study was drawn from the Odense University Pharmacoepidemiological Database (OPED).

In brief, it is a research database with full coverage of all reimbursed prescriptions in the Region of Southern Denmark (1.2 million inhabitants). The data included in each prescription record includes the prescription holder, the prescriber, the pharmacy, the date of dispensing and a full account of the dispensed product, including substance, brand name, route of administration, ATC-code and Defined Daily Dose (DDD) [START_REF] Gaist | The Danish prescription registries[END_REF]. Some drugs are completely exempt from re-imbursement and thus not covered by the database, including benzodiazepines, oral contraceptives, laxatives and certain antibiotics. Drugs with any degree of copayment are covered by the database.

All prescription redeemed by citizens of the Region of Southern Denmark (population 1.2 million) during 2009 were eligible for the analysis.

Analysis

We restricted the analysis to individuals who had redeemed 10 or more prescriptions during 2009. For each individual, we identified the prescriber who occurred most frequently on that individual's prescription list.

We defined the prescriber fidelity coefficient, FC Presc , as the proportion of an individual's redeemed prescriptions that were issued by the most frequent prescriber for that individual. Similarly, we defined the pharmacy fidelity coefficient, FC Pharm , as the proportion of an individual's prescriptions that were redeemed at the most used pharmacy. Unless otherwise stated, the FC Pharm and FC Presc are interpreted as a characteristic of a person. E.g., when calculating the average FC Pharm , we have calculated the average value for FC Pharm for all individual subjects in the study.

The FC Presc and FC Pharm are presented using standard descriptive statistics. Furthermore, we explored the dependency of FC Presc and FC Pharm on such variables as age, gender, number of prescriptions, whether the most frequent prescriber was a GP, whether the main pharmacy had more than one dispensing site and whether the most used pharmacy was urban. We defined urban pharmacies as those that were located in the Odense or Esbjerg municipalities (186,000 and 115,000 inhabitants) or who shared zip-code with another pharmacy. We analysed these associations by two linear regression models, one with FC Presc as dependant variable and one with FC Pharm as dependant variable. We excluded from this part of the analysis all subjects who had two or more pharmacies sharing position as the preferred and where at least one was near a competitor and at least one was not (N= 2246). Similarly, we excluded all subjects who had both a GP and a non-GP in a tied position among the preferred prescribers (N= 1779).

Finally we tabulated the proportion of prescriptions that were either issued by a non-main prescriber or redeemed at a non-main pharmacy within major drug classes. We grouped the drug classes according to the third level of the ATC code (e.g. M01A = NSAIDs). Only groups with more than 50,000 prescriptions (covering 88.7 % of the data) were reported.

Finally, we determined the proportion of prescription that were issued by a non-main prescriber or redeemed at a non-main pharmacy as a function of the month, thereby constructing a seasonality curve for FC Presc and FC Pharm .

Results

We extracted 10,067,798 prescriptions issued to 853,217 different individuals from the Region of Southern Denmark in 2009. After restriction to subject with 10 or more prescriptions, we had 8,246,064 prescriptions issued to 283.388 individuals. Of these subjects 121,734 (42.8%) were men, and their median age was 64 years (interquartile range 52 -75).

The average FC Presc was 0.882 (SD 0.158) and the average FC Pharm was 0.927 (SD 0.139). The average difference was 0.0446 (95% CI 0.0439-0.0453). 116,918 persons (41.2%) had an FC Presc of 1.00 and 182,030 (64.2%) had an FC Pharm of 1.00. Of those, 91,665 (32.3%) had a value of 1.00 for both parameters. 126,585 persons (44.7%) had a higher FC Pharm than FC Presc , while 50,640 persons (17.7%) had the opposite pattern.

There were 1,683 unique main prescribers and 242 unique main pharmacies.

Among the variables that were significantly associated with high FC Pharm , we found high age, male gender, high volume of prescriptions, main pharmacy having more than one dispensing site and use of a pharmacy with no competing pharmacies nearby (Table 1). Use of a pharmacy near a competitor was associated with a 0.053 lower FC Pharm than use of other pharmacies. When we restricted the analysis to only pharmacies near a competitor, the crude FC Pharm was 0.894. The variables that were associated with high FC Presc was high age, male gender, high number of prescriptions and use of a general practitioner as main prescriber (table 2). The dependency of FC Pharm and FC Presc on age and sex is shown in figure 1. In figure 2, we have shown the seasonality of both measures.

The major drug classes that were most often prescribed by a non-main prescriber were beta-lactams, antidepressants and opioids. Similarly, the major drug classes associated with use of non-main pharmacy were beta-lactams, antidepressants and adrenergics (inhalants), see table 3.

Discussion

The average FC Pharm was 0.927 and the average FC Presc was 0.882. Thus there is a slightly higher fidelity towards the pharmacy than towards the main prescriber. However, both have the possibility to have an excellent overview of their clients' medication. The actual overview also depends on factors such as the structure and the interface of the IT-solutions used by the prescriber and the pharmacies and the training of the prescribers and pharmacist. Also, our analysis is based on the actual dispensing of drugs, and as such we have no means of assessing to which extent the main prescriber is made aware of the prescriptions issued by specialists or other doctors to his patients.

The primary strength of the study is the high internal validity due to a high quality of the prescription data [START_REF] Gaist | The Danish prescription registries[END_REF]. Furthermore there is little selection bias, since all residents of Region of Southern Denmark were included in the analysis.

The primary weakness of the study is that the FC Pharm and the FC Presc are to a large extent determined by the underlying healthcare structure. Our result may thus not necessarily apply equally to other setting.

There are several factors in our setting that would favor a high FC Pharm over the FC Presc . First of all the pharmacies in Denmark are large units, often covering a substantial area, especially in comparison with the average pharmacy as seen in e.g. southern Europe. In our region there are 56 community pharmacies corresponding to a density of 1 pharmacy per 21.400 citizens. It is noteworthy however that the FC Pharm only shows a minor dependency on having multiple pharmacies nearby (table 1). Furthermore many doctors are specialists and thus only maintain a minor part of a patient's total medication. Other factors favour the FC Presc over the FC Pharm . The pharmacies are completely liberalised in Denmark, allowing patients to choose freely between pharmacies. On the other hand, each citizen is assigned a regular general practitioner that serves as a gate keeper, meaning that all medical contact, excluding emergencies, should go through the assigned general practitioner. Although it is possible to change general practitioner, this happens relatively rarely. Furthermore, there is a tendency in Denmark also among general practitioners to form larger units consisting of several practitioners under the same roof and using the same prescriber identifier. As such the single prescriber ID in our analysis can cover more than one individual prescriber. As these prescribers can see each other's prescribing to the individual patient within the group practice , they have the opportunity to avoid the problems that relate to multiple prescribers. Furthermore repeat prescriptions are registered as multiple single prescriptions in our analyses, even though they only represent a single prescription decision. By definition, repeat prescriptions are issued by the same prescriber, but not necessarily redeemed at the same pharmacy, again a factor that would favour a high FC Presc relatively to the FC Pharm . Finally, general practitioners frequently take over the prescribing of specialised drug regimes as soon as the medication is stable. In conclusion the fidelity coefficient is highly dependent on the health care structure. Most of the factors in our setting point towards a higher FC Presc . than FC pharm . It is therefore surprising to find a FC Pharm greater than the FC Presc .

Our analysis in table 3 shows that antibiotics account for most of the infidel prescriptions, which hardly comes as a surprise. It is more interesting to find the groups of 'antidepressants', 'antipsychotics' and 'antithrombotic agents' being so highly represented. These three groups are known to often represent long term treatments and also show a wide range of possibly dangerous drug-drug-interactions, especially regarding the antithrombotic agents [START_REF] Hallas | Use of single and combined antithrombotic therapy and risk of serious upper gastrointestinal bleeding: population based case-control study[END_REF]. Combining the numbers for these three groups show that while 111,733 of these prescriptions where made by others than the most used prescriber only 57,671 where redeemed away from the most used pharmacy. While both numbers are higher than desired, this emphasizes the central role of the pharmacy in discovering and preventing drug-drug-interactions.

The importance ofthe Fidelity Coefficient for monitoring medication profiles, avoiding doubling or interations is most obvious in a setting where a data on the medication of the single individual are not readily available for the health service practitioner. This is still the case in most countries. In Denmark, each redemption of a prescription is registered but no complete list of 'current treatment' is produced for routine care. This will probably change in the coming years, with new IT-solutions.

Several questions arise from this study. First, it would be interesting to explore how the 'fidelity coefficient' differs across different populations and different health care models. It might even be possible, through subsequent studies, to link the 'fidelity coefficient' to other parameters such as ADE-rates on a population scale. Lastly the 'fidelity coefficient' could be used as a tool to refine future population based analyses, for example by having a high fidelity as an exclusion or inclusion criteria.

Table 1 : The dependency of the pharmacy fidelity coefficient on explanatory variables.

 1

	Base FC Pharm	0.838 [0.836 ; 0.839]
	Age*	0.018 [0.017 ; 0.018]
	Male gender	0.011 [0.010 ; 0.012]
	Number of prescriptions †	0.003 [0.003 ; 0.003]
	Main pharmacy near competing pharmacy ‡	-0.053 [-0.053 ; -0.052]
	Main pharmacy having more than one dispensing site	0.006 [0.005 ; 0.007]

* The

influence of age over FC Pharm is given as the change per 10 years. † The influence of number of prescriptions over FC Pharm is given as the change per 10 prescriptions. ‡ The classification of 'nearby pharmacies' is given in the method section.

Table 2 : The dependency of the prescriber fidelity coefficient on explanatory variables.

 2

	Base FC Presc	0.627 [0.625 ; 0.630]
	Main prescriber being a general practitioner	0.200 [0.198 ; 0.202]
	Age*	0.012 [0.011 ; 0.012]
	Male gender	0.002 [0.001 ; 0.003]
	Number of prescriptions †	0.002 [0.002 ; 0.002]

* The influence of age over FC Pharm is given as the change per 10 years. † The influence of number of prescriptions over FC Pharm is given as the change per 10 prescriptions.

Legend to figure 1: The dependency of the Fidelity Coefficient on age and sex

Legend to figure 2 Season variability of the Fidelity Coefficient


A Zouaghi

M Bellet

Y Bienvenu

G Perrin

D Cedat

M Bernacki

Modélisation de la phase de compaction du procédé CIC à l'échelle mésoscopique

Keywords: CIC, Compaction de poudre, fonction level-set, technique de remaillage, échelle mésoscopique

La Compaction Isostatique à Chaud (CIC ou Hot Isostatic Pressing HIP en anglais) est un procédé de compaction de poudre qui consiste à soumettre une enveloppe remplie de poudre à un traitement thermique sous pression isostatique. Dans ce procédé, la déformation viscoplastique des particules est un mécanisme prépondérant. Cependant, la compréhension de l'impact global des conditions aux limites thermo-mécaniques et de la granulométrie de poudre sur la microstructure obtenue après ce mécanisme n'est pas simple. Une méthode éléments finis basée sur l'utilisation de level-sets couplée avec une technique de remaillage est proposée afin de modéliser la déformation viscoplastique des particules de poudre à l'échelle mésoscopique sur un Volume Elémentaire Représentatif (VER). La méthode consiste à générer, dans un maillage éléments finis, un empilement de particules sphériques représenté implicitement par des fonctions level-set. Afin d'éviter un coût de calcul important, une adaptation de maillage est également effectuée au niveau des interfaces des particules pour les décrire correctement et gérer la forte discontinuité des propriétés physiques entre les deux sous-domaines (particules et espace inter-particulaire). Les résultats de ce travail sont présentés et discutés.

Introduction

En raison des possibilités qu'elle offre à la fois d'un point de vue technique mais également économique, le procédé de Compaction Isostatique à Chaud est de plus en plus utilisé dans les industries Oil and Gas (offshore) et aéronautique. Cette technique permet de produire des pièces métalliques ou céramiques haut de gamme. En effet, ce procédé permet d'obtenir des pièces densifiées de forme plus ou moins complexe et de microstructure uniforme. La CIC consiste à soumettre un conteneur rempli de poudre à un traitement thermique sous pression isostatique appliquée par l'intermédiaire d'un gaz inerte. La modélisation du procédé CIC représente un projet de recherche à la fois ouvert et complexe vu qu'il met en jeu un grand nombre de phénomènes physiques différents. Le mécanisme qui intervient en premier est la déformation viscoplastique des particules. Ensuite, suite à l'évolution de la pression et de la température au cours du temps, plusieurs mécanismes de transport par diffusion apparaissent et deviennent prépondérants [START_REF] Helle | Hot-isostatic pressing diagrams: New developments[END_REF].

Aujourd'hui, grâce aux progrès considérables des moyens de calculs mais aussi expérimentaux, nous pouvons avoir accès à l'information microscopique et la modéliser. La modélisation à l'échelle d'un volume élémentaire représentatif (VER) permet de mettre en oeuvre les différents phénomènes microstructuraux intervenant. Le travail à cette échelle permet d'apporter des éléments de compréhension sur les caractéristiques et la prédiction du comportement des pièces fabriquées et d'étudier l'impact des paramètres du procédé (température, pression, durée des essais, granulométrie,…) sur les caractéristiques finales. Les principaux objectifs de ce travail consistent à modéliser le mécanisme prépondérant en début de CIC, soit celui de la déformation viscoplastique des particules, et à suivre l'évolution de la cinétique de densification à l'échelle d'un VER.

Dans cet article, nous commencerons par présenter les équations du problème mécanique ainsi que les lois de comportement adoptées et les conditions aux limites appliquées. Ensuite, nous présenterons l'approche numérique retenue. Enfin, une dernière partie est dédiée à la présentation et à la discussion des résultats obtenus.

Modélisation mécanique

Pour modéliser le procédé CIC à l'échelle mésoscopique, nous avons considéré un Volume Elémentaire Représentatif (VER) en 2D composé de particules de poudre sphériques (matière) et d'un milieu interparticulaire (MI). Le suivi de la déformation des particules représente un des objectifs majeurs de notre modélisation. Ainsi, pour ne pas être obligé de suivre l'évolution du MI aux extrémités libres du domaine, nous avons adopté une description eulérienne.

Equations générales de la mécanique

Le modèle mécanique est gouverné par l'équation fondamentale de la dynamique et l'équation de conservation de la masse suivantes :

g dt dv ρ σ ρ + ⋅ ∇ = ; 0 ) ( = ⋅ ∇ + ∂ ∂ v t ρ ρ , (1)
où ν désigne la vitesse, σ le tenseur des contraintes, ρ la masse volumique et g l'accélération gravitationnelle.

Lois de comportement

Le matériau objet de l'étude est un acier inox 316L. La modélisation du mécanisme de déformation viscoplastique des particules est basée sur une loi de type puissance [START_REF] Helle | Hot-isostatic pressing diagrams: New developments[END_REF][2][3], à savoir :

ε ε 1 ) 3 ( 2 - = m K S , (2)
où S, et ̅ représentent respectivement, la partie déviatoire du tenseur des contraintes, la vitesse de déformation et la vitesse de déformation équivalente au sens de Von Mises ( ε ε ε

: 3 2 =
). K représente la consistance du matériau et m la sensibilité de la contrainte d'écoulement à la vitesse de déformation. K et n sont considérés dépendants de la température [START_REF] Bouaziz | Creation of a material data file for modelling of an austenitic stainless steel[END_REF] [START_REF] Kim | Densification behavior of 316L stainless steel powder under high temperature[END_REF]. Concernant le MI, nous lui attribuons le comportement d'un fluide newtonien (K=η et m=1 dans l'équation (2)), à savoir :

) ( 2 ε η dev S = , (3)
avec η la viscosité dynamique.

Coefficient de compressibilité

Afin de mieux approcher le comportement du MI (vide ou quasi-vide en CIC), nous avons résolu les équations de Navier Stokes compressible. Ces dernières sont obtenues en introduisant un terme (C k ) dans l'équation de conservation de la masse :

g p S v v t v ρ ρ + ∇ - ⋅ ∇ = ⋅ ∇ + ∂ ∂ ) ( , (4) 0
= + ⋅ ∇ p C v K . ( 5
)
Pour introduire l'incompressibilité des particules, nous avons adopté des valeurs du coefficient de compressibilité C K =ε' très proches de 0. Concernant le MI, les valeurs de C k sont assez importante (C k >>ε') pour tenir compte de la compressibilité du milieu.

Modélisation numérique

Les simulations numériques présentées ont été réalisées avec la librairie éléments finis C++ ''CimLib'' développée au CEMEF.

Méthode level set

Le VER sujet de ce travail est un domaine fixe dans lequel les interfaces entre les différents sousdomaines (particules de poudre, milieu englobant et MI) ne sont connues qu'implicitement à travers les valeurs d'une fonction définie sur tout le domaine de calcul. Cette fonction est une fonction distance signée à l'interface d'intérêt dans le domaine de calcul . La level set α est choisie positive à l'intérieur du sous-domaine (noté ω i ) et négative partout ailleurs. Ainsi, l'isovaleur zéro i ω ∂ représente l'interface que l'on cherche à déterminer. Cette fonction est définie par la fonction suivante :

i i i i x si x d x x si x d x ω ω α ω ω α - Ω ∈ ∂ - = ∈ ∂ = ) , ( ) ( ; ) , ( ) ( . (6)
Dans un premier temps, vu que toutes les particules ont le même comportement et afin d'éviter un coût de calcul très important, nous avons travaillé avec une seule fonction level-set globale pour toutes les particules. En supposant que chaque particule possède une fonction distance

) (x i α définissant son interface, la fonction )) ( ( max ) ( x x i i α α =
représente la fonction level set globale qui définit l'ensemble de toutes les particules de poudre. En effet, son isovaleur zéro correspond à l'enveloppe de la réunion de toutes les particules. Dans la bibliothèque CimLib, une deuxième fonction level set tronquée 'auto-déterminante' peut être utilisée [START_REF] Coupez | Approche monolithique pour l'IFS[END_REF]. Cette fonction permet de ne prendre en compte la fonction level-set qu'aux alentours des interfaces, tout en gardant une bonne régularité. Dans la bibliothèque CimLib, plusieurs solveurs ont été construits pour transporter l'ensemble des lignes de niveau [START_REF] Coupez | Approche monolithique pour l'IFS[END_REF].

Lois de mélange

Les variables ρ, C K , K et m présentées précédemment sont des variables moyennées par l'intermédiaire d'une fonction de mélange. En supposant que nous avons 2 fonctions level-set : la première caractérisant l'interface des particules et la deuxième les interfaces du milieu englobant, alors les lois de mélange de ces différents paramètres peuvent être présentées sous la forme suivante :

MEnglobant MI par M M M M ) ( f )) ( f - ](1 )) ( f - (1 ) ( f [ 2 2 1 1 α α α α + + = , (7)
= ) (α f 1 pour e > α ; = ) (α f             + + e e α π π α sin 1 1 2 1 pour e < α ; = ) (α f 0 pour e - < α , (8)
avec e épaisseur du mélange souhaitée choisis égale à 2*h mesh .

Adaptation de maillage

La précision de la méthode des éléments finis dépend de la taille des éléments. Plus ces éléments sont petits plus le calcul est précis et plus les simulations et les résultats seront fiables. Pour des raisons évidentes de coût de calcul, il n'est pas possible de raffiner trop uniformément la taille des éléments. Mais, il est possible d'optimiser le maillage en raffinant seulement dans les zones ''utiles'' et en dé-raffinant en dehors de celles-ci. Ceci nous amène au problème de remaillage adaptatif. Au cours de ce travail, nous avons utilisé une méthode prédéterminée de génération de maillage dans laquelle nous choisissons dès le début d'adapter le maillage de manière anisotrope ou isotrope sur une certaine épaisseur autour de l'isovaleur 0 des différents sous-domaines [7].

Applications et résultats

Deux types de conditions aux limites (CL) ont été imposés sur un milieu englobant permettant de les appliquer sur l'ensemble particules-MI. Dans une première approche, une pression isotrope a été appliquée (Fig. 2). Dans une deuxième approche, nous avons appliqué des contraintes normales sur deux milieux englobants, et en imposant des conditions de symétrie sur les deux autres frontières du VER (les contraintes sont appliquées de manière faible dans la formulation du problème mécanique). Ces conditions aux limites permettent de mieux s'approcher de la réalité du couplage multi-échelle (Fig. 1). Les paramètres physiques et numériques utilisés sont présentés dans le tableau 1.

Dans les cas développés, les particules se déforment au cours du temps et les cous (surface de contact entre deux particules adjacentes) entre les différentes particules et la densité relative augmentent. Nous avons étudié, dans ce travail, l'influence des paramètres numériques sur la conservation du volume des particules. Ces paramètres sont notamment: le coefficient de compressibilité, la période de remaillage, les lois de mélange et le nombre de noeuds. Nous avons constaté que le coefficient de compressibilité a une grande influence sur la conservation de matière. Un coefficient C k égal à zéro serait idéal pour minimiser la perte de volume. Cependant, il est nécessaire de ne pas imposer une loi de mélange trop raide. Ainsi nous avons appliqué des coefficients de l'ordre de 10 -10 Pa -1 . Concernant la période de remaillage, il est souhaitable d'augmenter ce paramètre afin de ne remailler que s'il est vraiment nécessaire (lorsque les interfaces sont sur le point de sortir de l'épaisseur du maillage fin), afin de limiter le coût de calcul ainsi que la diffusion numérique dûe au transport des variables lors du remaillage.

Etant donné ce formalisme initial, plusieurs difficultés numériques sont apparues : Déraffinement du maillage au niveau des cous. En effet, le fait de travailler avec la fonction distance globale entraîne la disparition de la description des cous entre particules puisqu'elle ne permet de décrire que l'enveloppe de la réunion des particules. L'adaptation de maillage est réalisée au niveau de l'isovaleur zéro de cette fonction. Une méthode de coloration de graphe a alors été utilisée afin de former des familles de particules de sorte que deux particules adjacentes et en contact soient associées à deux fonctions level-set distinctes. Cette méthode permet de conserver la description des cous entre particules tout en limitant très fortement le nombre de fonctions level-set utilisées par rapport à l'utilisation d'une fonction level-set par particule.

La prise en compte d'une loi de comportement de type puissance pour la modélisation du comportement des particules, sur laquelle est basée la modélisation du premier mécanisme prépondérant au début de la CIC « déformation viscoplastique des particules » rend le problème mécanique à résoudre plus raide.

Le contrôle de la conservation de matière représente un des points importants. L'influence des différents paramètres sur cet observable a été étudiée. Une perte de matière de l'ordre de 2% a été remarquée.

Dans une deuxième approche, un passage à des VER de forme circulaire en 2D a représenté le sujet d'un travail effectué pour des cas isothermes (FIG. 2) mais aussi pour un cycle allant de 800 à 1150°C en une heure (proche de la dernière heure de la montée en pression et température dans un cycle CIC). Les points essentiels ressortant de l'ensemble de cette modélisation numérique à l'échelle mésoscopique du procédé CIC sont: un bon profil d'évolution de la densité relative (FIG. 3), une évolution remarquable des particules de poudre surtout dans le centre du VER et une perte de matière optimisée de l'ordre de 2%.

Conclusion

Ce travail a permis de mettre en place une procédure de calcul qui permet de modéliser la compaction d'un ensemble de particules de poudre en 2D et d'interpréter l'évolution de la densité relative résultante. Les travaux de modélisation des différents sous domaine du VER (particules et MI) ont été basés sur la prise en compte d'un comportement de type loi de puissance et la résolution des équations de Navier Stokes compressible. Le modèle numérique retenu est basé sur un domaine fixe dans lequel les interfaces entre les différents sous-domaines ne sont connues qu'implicitement à travers les valeurs d'une fonction distance définie sur l'ensemble du domaine de calcul (approche « level-set »). Les propriétés physiques des différents domaines ont été introduites à travers des lois de mélange. Notre travail futur portera sur l'amélioration des résultats obtenus (conservation de matière et optimisation des paramètres numériques). De plus, une comparaison avec les résultats analytiques existants [START_REF] Atkinson | Fundamental aspects of hot isostatic pressing: an overview[END_REF][1] pour un plus grand nombre de particules est envisagée. Nous nous intéresserons également à la compaction d'un volume élémentaire représentatif représentant statistiquement des microstructures réelles en termes de densité, tailles, nombres et morphologie de particules en 2D et 3D. Enfin, les résultats obtenus permettront d'améliorer les modèles macroscopiques traditionnellement utilisés.

  Le choix des fonctions caractéristiques ) ( f 1 α et ) ( f 2 α est très important vu qu'il conditionne la transition des caractéristiques physiques des différents sous domaines du VER. Afin d'avoir une transition des propriétés plus régulière, nous avons considéré des fonctions caractéristiques définies par la relation suivante :

FIG 1 .

 1 FIG 1. Evolution d'un VER carré en 2D (a) état initial (b) après 20% et (c) après 40% de ''déformation'' (réduction relative de la surface de VER) (d) norme de la vitesse après 40% de déformation

FIG 3 .

 3 FIG 3. Evolution de la densité relative pour un cycle allant de 800 à 1150°C en une heure


Arghya Majee

Alois Würger

Charging of heated colloidal particles using the electrolyte Seebeck effect

Keywords:

Charging of heated colloidal particles using the electrolyte Seebeck effect

Arghya Majee, Alois Würger

Selective transport and controlled pattern formation are of fundamental interest in microfluidics and biotechnology [1]. Particle focussing devices [2][3][4] and macromolecular traps [5][6][7] have been designed by applying chemical or thermal gradients. In "active colloids" there is no external symmetry breaking field: Thermodynamic forces arise from an embarked chemical reactor [8,9] or from non-uniform laser heating of Janus particles [10]. In both cases the colloid self-propels in an anisotropic environment that is created by the concentration or temperature variation along its surface. The interplay of selfpropulsion and Brownian motion leads to a complex diffusion behavior [8][9][10][11].

Locally modifying material properties by heating a single nanoparticle or molecule in a focussed laser beam is by now a standard technique. The temperature dependent refractive index was used for the photothermal detection of a single non-fluorescent chromophore [12]. Heating a spherical nanoparticle induces a radial temperature profile in the surrounding fluid and, because of the viscosity change, an enhancement of the Einstein coefficient [13]. The non-uniform laser absorption of half-metal coated particles leads to a temperature variation along its surface; the resulting self-propulsion adds a ballistic velocity component and thus increases the effective mean-square displacement [10].

In this Letter, we point out that heating confers a net charge on colloidal particles, and how this electrolyte Seebeck effect in the vicinity of a heated bead can be used for selective transport and controlled interactions. After a brief reminder of the well-known case of a constant temperature gradient, we derive the expressions for the thermocharge and the electric field induced by a spherical particle with excess temperature δT . As possible applications, we then discuss the thermoelectric pair potential, aggregation or depletion of a molecular solute in the vicinity of a hot particle, and colloidal separation through velocity differentiation.

The response of a salt solution to a constant thermal gradient ∇T is illustrated in the left panel of Fig. 1. Because of their temperature dependent solvation energy, positive and negative salt ions migrate along the gradient. In general one of the species moves more rapidly, thermoelectric field E FIG. 1: Left panel: Seebeck effect in an electrolyte solution that is cooled at the left side and heated at the right. We show the case of a positive Seebeck coefficient S, where cations and anions accumulate at the cold and hot boundaries, respectively. These charged layers have a thickness of about one Debye length (dashed lines). The corresponding thermoelectric field E is constant in the bulk and vanishes at the boundaries [20]. Right panel: Seebeck effect in the vicinity of a hot particle with excess temperature δT . Due to the radial temperature gradient, a net charge Q accumulates within one Debye length from the particle surface. The charge density and the radial electric field are shown in Fig. 2 below. The counterions are at the vessel boundary.

resulting in a thermopotential between the cold and hot boundaries of the sample [14] and a macroscopic electric field E = S∇T , which is proportional to the thermal gradient and to the Seebeck coefficient S [15]. The bulk solution is neutral; yet opposite charges accumulate at the boundaries and screen the electric field in a layer of one Debye length. In the last years it has become clear that charged colloids in a temperature gradient are a sensitive probe to the Seebeck effect of the electrolyte solution [16,17]: The field E and thus the colloidal velocity depend strongly on the salt composition and are particularly important in the presence of molecular ions containing hydrogen [18][19][20]; this thermo-electrophoretic driving has been confirmed for SDS micelles in a NaCl 1-x OH x solution, where a change of sign of the drift velocity has been observed upon varying the parameter x [17].

Thermocharge of a hot colloid. Now we consider the Seebeck effect in the vicinity of a heated particle, as shown in the right panel of Fig. 1. The qualitative features are readily obtained by wrapping the hot boundary of the one-dimensional case (left panel) onto a sphere of radius a. Its excess temperature δT results in a thermal gradient

∇T = - δT a r 2 (1)
and, at distances well beyond the Debye length, in a thermoelectric field E = S∇T . Its complete expression, in particular close to the particle surface, is obtained from the stationary electro-osmotic equations for the ion currents

J ± = -D ± ∇n ± ∓ n ± eE k B T + 2n ± α ± ∇T T , (2)
which comprise normal diffusion with coefficients D ± , electrophoresis with the Hückel mobility for monovalent ions, and thermal diffusion with parameters α ± . The latter are reduced values of the ionic Soret coefficient, introduced by Eastman as a measure for the electromotive force of an electrolyte thermocouple [14]; experimental values are found in Refs. [15,21,22]. The thermoelectric field E and the charge density ρ = e(n + -n -) are obtained from the steady-state condition J ± = 0 and Gauss' law divE = ρ/ε. Linearizing the currents in the small gradients and solving the coupled differential equations for E and ρ, we obtain the radial steady-state thermoelectric field

E = S∇T 1 - r + λ a + λ e (a-r)/λ , (3)
with the Debye length λ and the Seebeck coefficient S = (α + -α -)k B /e. For a detailed calculation see Ref.

[23]. With (1) the field E is zero on the particle surface and at infinity, as required by electrostatic boundary conditions. At distances well beyond λ the exponential factor vanishes; the remaining long-range contribution E = S∇T varies with the inverse square of the distance r. From Gauss' law one obtains the charge density [23]

ρ = Qe (a-r)/λ 4π(a + λ)λr ,
which is concentrated within about one Debye length from the particle surface. Fig. 2 illustrates the variation of E and ρ with distance for different values of λ; the former is long-range whereas the latter decays exponentially. The net thermocharge carried by an otherwise non-ionic particle, depends on its radius in units of the Bjerrum length ℓ B = 7 Å, the ratio of excess and absolute temperature, and the reduced Seebeck parameter α + -α - [19].

Q = -e(α + -α -) a ℓ B δT T , (4)
In physical terms the charge Q arises from the difference in thermo-osmotic pressure of positive and negative ions, which in turn is related to the ionic Soret parameters α ± /T [14]. For α + > α -the anions show thermal diffusion toward higher temperature, thus accumulating a negative charge at the particle surface. The corresponding cations are located at r → ∞; in the case of a spherical sample container of radius R, the corresponding surface charge density Q/4πR 2 is very small. Numerical values for the Seebeck coefficient of several electrolytes are given in Table 1. For small ions the numbers α ± are of the order of unity; higher values occur for molecules containing hydrogen. For a 100 nm-bead in NaOH or HCl solution with δT = 30 K, one finds that Q corresponds to about 40 elementary charges; still higher [15,21], and for tetrabutylammonium nitrate (TBAN) in water (w) and dodecanol (d) [22]. For comparison, S of simple metals is of the order of a few µV/K. The Seebeck coefficient is related to Eastman's ionic entropy of transfer 2kBα± through S = (kB/e)(α+ -α-) [19]. Experimental values for various ions are given in Refs. [14,15,21,22 values occur for protonated salts. Equations ( 3) and ( 4) are the main formal results of this paper. In the remainder we discuss how the thermocharge allows to actuate colloidal motility and interactions, and how the thermoelectric field can be used for locally accumulating or depleting an additional charged molecular solute. As an overall feature we estimate the thermoelectric response time. Because of the fast equilibration of heat flow and temperature, thermocharging occurs on the time scale of thermal diffusion of salt ions over one Debye length. With the above parameters one finds a relaxation time of the order of microseconds. Thus on the scale of colloidal motion, thermocharging is an almost instantaneous process.

Colloid-colloid forces. We start with the electric force QE between two hot particles at a distance R. Assuming R ≫ λ and using the definition of the Bjerrum length, we find

F = Q 2 4πεR 2 .
(

) 5
Thus in an electrolyte with finite Seebeck coefficient, heating disperses colloidal aggregates and strongly affects collective effects due to thermophoretic or hydrodynamic interactions [24]. So far we have considered non-ionic colloids. A particle carrying a proper charge Q p gives rise to an additional electric field E p = Q p e -(r-a)/λ /(4πελr).

Depending on the sign of Q and Q p , the superposition E + E p shows a complex spatial variation; note that E p is screened whereas E is not. Thermo-electrophoresis. Thermocharging provides a unique tool for creating a radial electric field in an electrolyte solution. For a micron-size bead with an excess temperature δT = 30 K, the field E may attain 10 4 V/m in its immediate vicinity, and a few V/m at a distance of 100 microns. The electrophoretic velocity of a molecular solute with zeta potential ζ,

u = 2 3 εζ η E, (6)
varies between 10 µm/s and 10 nm/s. Depending on the sign of the zeta potential ζ and of the Seebeck coefficient, molecular ions are attracted or repelled by the thermocharge. As illustrated in Fig. 3a., this can be used

E ext a) δT > 0 δT = 0 b) E ext c)
FIG. 3: Pattern formation and selective transport due to thermocharging. a) The thermoelectric field of a hot particle induces electrophoretic motion (6) of a charged molecular solute. Depending on the sign of the electrolyte Seebeck coefficient S and the molecular zeta potential ζ, the colloidal thermocharge results in depletion or accumulation of the solute. b) In an external electric field Eext , heated particles with excess temperature δT and thermocharge Q, acquire a size-dependent velocity (7). c) Because of their different optical absorption properties, metallic and semiconducting carbon nanotubes differ in their thermocharge and in their response to an electric field.

for accumulating or depleting a molecular solute in the vicinity of the particle.

More complex patterns are realized by superposing E with the screened field E p of a proper charge. In addition to thermo-electrophoresis (6), the radial temperature profile results in thermal diffusion of the solute molecules, due to both double-layer and dispersion forces [19,25]. Finally we mention that the thermoelectric field E of Janus particles comprises a strongly anisotropic short-range component.

Selective transport. Sorting molecular or colloidal solutes by size is of interest for various applications. The sedimentation potential being rather ineffective for submicron particles, common methods are based on electrophoresis or on motion driven by thermodynamic forces. Since the free-solution mobilities are in general independent of size and molecular weight, velocity differentiation is achieved only after adding a molecular solute as in gel electrophoresis [26,27], or by spatial flow or field modulation [28][29][30].

Here we show that thermocharging in the presence of an applied electric field E ext , provides an efficient means for separating particles by size. The force density ρE ext exerted by the external field on the charged fluid results in a drift velocity u ext of the particle; solving the stationary Stokes equation one finds [23]

u ext = QE ext 6πη(a + λ) . (7)
Note the explicit dependence on the particle size, in contrast to the Helmholtz-Smoluchowski mobility. In view of the thermocharge (4), the most interesting dependencies arise from the excess temperature δT . Assuming a constant volume absorption coefficient β, one finds that the excess temperature varies with the square of the radius,

δT = a 2 βI 3κ , ( 8
)
where I is the laser intensity and κ the thermal conductivity of the solvent.

According to (8) the excess temperture increases with the square of the bead size; thus the drift velocity u ext varies with the particle surface in the Hückel limit (a < λ) and with its volume in the case a > λ, as illustrated in Fig. 3b. As an order-of-magnitude estimate, heating the beads by δT = 30 K and applying a field E app ∼ 10 4 V/m results in a velocity of about 10 µm/s. The above argument holds true for non-spherical solute particles, albeit with different geometrical factors. The excess temperature of metal-coated polystyrene beads is linear in the radius. For polymers the charge Q is proportional to the chain length or number N of monomers, whereas the friction coefficient varies with the gyration radius R ∝ N ν , resulting in a velocity u ∝ N 1-ν . In aqueous solution most colloids carry a proper charge with surface potential ζ p , resulting in an additional velocity u p = (εζ p /η)E ext . Still, the thermocharge leads to a significant dispersion of the total velocity u ext + u p .

We conclude with a possible application to the separation of carbon nanotubes by their wrapping structure [29,31]. The electronic and optical properties of single-wall nanotubes depend crucially on their "chiral vector" (n, m) that describes the orientation of the graphene structure with respect to the tube axis. Depending on these indices, one has metallic or semiconducting nanotubes with a characteristic bandstructure and a particular optical spectrum. The excess temperature δT = a βI/κ of a nanotube depends on its radius a and the absorption per unit area β of its graphene sheet, and so does the drift velocity u ext . As illustrated in Fig. 3c, by chosing an appropriate laser wave length, one could selectively heat nanotubes with a given chiral vector and separate them through thermocharge electrophoresis.

FIG. 2 :

 2 FIG. 2:Thermoelectric field E and charge density ρ as a function of the distance r from the particle centre for different values of the Debye length λ and fixed particle size a = 1µm. We have used the parameters α+ -α-= -10 and δT = 30 K. The dashed line gives the bulk law E = S∇T .

TABLE I :

 I Seebeck coefficient S for NaCl, HCl and NaOH in aqueous solution

  ].

	Salt/solvent NaCl/w NaOH/w HCl/w TBAN/w TBAN/d
	S (mV/K)	0.05	-0.22	0.21	1.0	7.2
	α + -α -	0.6	-2.7	2.6	12	86